# Solving Right Triangle in Trigonometry

Solving right triangle in trigonometry is really easy. We just need to remember some mnemonics and a little background in Pythagorean Theorem.

This topic is very powerful in many ways. Right triangle has a very huge application in both basic and advanced mathematics. The foundation in solving this easily is to use SOH-CAH-TOA formula.

Given the right triangle,

Right Triangle in Trigonometry

Parts:

$\Theta$ – this is the angle itself

Adjacent Side (A) – this is the side adjacent to the angle, so if $\theta$ is in $\angle C$, the adjacent side will be the side $\overline{BC}$

Opposite Side (O) – this is the side always opposite to the given angle

Hypotenuse (H) – this is constant part, the hypotenuse is always the longest side of the triangle opposite the right angle.

When solving this problem, we need to make sure the location of the angle. If it is on the 1st, 2nd, 3rd or 4th quadrant because the sign of cosine and sine varies depending on the location of the angle. Take a look of the photo below.

Signs of Sine and cosine in each quadrant

SOH-CAH-TOA mnemonics:

$sin\theta=\displaystyle\frac{O}{H}$

$cos\theta=\displaystyle\frac{A}{H}$

$tan\theta=\displaystyle\frac{O}{A}$

Worked Problem 1:

Given that $sin\theta =\displaystyle\frac{3}{5}$ and $\theta$ is in quadrant 2. Find the value of the following:

a. $cos\theta$

b. $tan\theta$

c. $sin2\theta$

Solution:

Draw a quick draft of the figure so that you can identify where is the adjacent and opposite sides.

Since we’re dealing with right triangle, we can always use Pythagorean Theorem to solve for the unknown side.

$H^2=A^2+O^2$

$5^2=A^2+3^2$

$A^2=25-9=16$

$A=\sqrt{16}=\pm4$

Solving for $cos\theta$: We use CAH, consider that cosine in quadrant 2 is negative

$cos\theta=\displaystyle\frac{A}{H}$

$cos\theta=\displaystyle\frac{-4}{5}$

Solving for $tan\theta$: In quadrant 2, sine is positive and cosine is negative

$tan\theta=\displaystyle\frac{O}{A}$

$tan\theta=\displaystyle\frac{3}{-4}=-\displaystyle\frac{3}{4}$

Solving for $sin2\theta$:

Now, we need to break up the a little $sin2\theta$

Recall that $sin2\theta=2sin\theta cos\theta$, from double angle identity.

$sin2\theta=2sin\theta cos\theta$

$sin2\theta=2\cdot \displaystyle\frac{3}{5}\cdot -\displaystyle\frac{4}{5}$

$sin2\theta=-\displaystyle\frac{24}{25}$

Worked Problem 2:

Find    $sin(x-y)$   if   $sin(x)=\displaystyle\frac{3}{5}$   and   $sin(y)=\displaystyle\frac{5}{13}$. If    $x,y$    are on the 1st quadrant.

Solution:

I know what you’re thinking, $sin(x-y)\ne sin(x)-sin(y)$. We cannot use the distributive property because we’re dealing with angles. Not algebraic quantities.

Since we have two angles x and y, we also need to create 2 triangles.

In figure 1, we need to complete the side of the triangle.

$H^2=A^2+O^2$

$5^2=3^2+O^2$

$A^2=5^2-3^2=16$

$A^2=\pm 4$

Same goes in figure 2,

$13^2=A^2+5^2$

$A^2=13^2-5^2$

$A^2=\pm 12$

Now, recall the sum and difference formula of $sin(x\pm y)=sin(x)cos(y)\pm cos(x)sin(y)$ (sincoscosin mnemonic)

Thus, $sin(x-y)=sin(x)cos(y)-cos(x)sin(y)$

Now, we don’t have   $cos(x)$   and   $cos(y)$,  so let’s look for it.

Solving for $cos(x)$:  Refer to figure 1, remember that location of the angle, since it is on quadrant 1, all values of A and O are positive.

$cos(x) =\displaystyle\frac{A}{H}$

$cos(x) =\displaystyle\frac{4}{5}$

Solving for   $cos(y)$: Refer to figure 2, again the angle is in quadrant 1 so the values of opposite and adjacent are all positive.

$cos(y) =\displaystyle\frac{A}{H}$

$cos(y) =\displaystyle\frac{12}{13}$

Going back to    $sin(x-y)$

$sin(x-y)=sin(x)cos(y)-cos(x)sin(y)$

$sin(x-y)=\displaystyle\frac{3}{5}\cdot\displaystyle\frac{12}{13}-\displaystyle\frac{4}{5}\cdot\displaystyle\frac{5}{13}$

$sin(x-y)=\displaystyle\frac{16}{65}$

To outline the steps and guide you to solve right triangle in trigonometry, here are some few tips and steps that you might want to consider so that you would where to start.

Step 1: DRAW the figure. Unless you have the power of wild imagination of the images, you can skip this, but if you’re in the learning process, always draw!

Step 2: Complete the parts of right triangle using Pythagorean Theorem. Easy right?

Step 3: Location of angle. The sign of your answer is dependent on the location of the angle. Mathematics is so strict; always take into account the sign after solving the problem. If your sign is wrong, you are as good as your 3-year old brother. Your answer is wrong.

Step 4: Give the answer to the question if what is being asked. Remember that, because you might be pre occupied with the series of steps and you have forgotten that the problem is just asking for the hypotenuse. Do not overthink!

Step 5: Be cool. The key to making your life easier in trigonometry is to memorize the basic trigonometric identities and learn how to use them properly. Trigonometric Identities is like knowing how to add, subtract, multiply, and divide in arithmetic.

So, you think you can do these simple steps?

### Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

### 234 Responses

1. Karen says:

I have been reading out some of your posts and i can state pretty nice stuff. I will make sure to bookmark your blog.

2. My spouse and i felt very thankful when Chris could finish up his basic research from your precious recommendations he had through the site. It is now and again perplexing to simply choose to be releasing helpful tips which often most people could have been selling. So we remember we have got the writer to thank for that. The specific explanations you have made, the straightforward site menu, the relationships you give support to foster – it is everything amazing, and it’s aiding our son and us reason why this subject is amusing, and that is seriously essential. Thank you for the whole thing!

3. this site says:

LCLGek Im thankful for the blog post.Really looking forward to read more. Awesome.

4. Way cool! Some extremely valid points! I appreciate you writing this post plus the rest of the site is extremely good.

5. Very neat article post.Really thank you! Really Cool.

6. Dramacool says:

7. It is almost not possible to find knowledgeable folks within this subject, on the other hand you sound like you realize what you are speaking about! Thanks

8. A round of applause for your article.Really thank you! Really Great.

9. Whats Taking place i am new to this, I stumbled upon this I have found It absolutely useful and it has helped me out loads. I am hoping to contribute & aid other customers like its aided me. Good job.

10. Thanks a lot for the post. Cool.

11. This blog was how do you say it? Relevant!! Finally I ave found something which helped me. Cheers!

12. It as actually a great and helpful piece of info. I am happy that you shared this useful information with us. Please stay us up to date like this. Thank you for sharing.

13. Thank you, I ave been searching for facts about this topic for ages and yours is the best I have found so far.

14. china dealz says:

Major thanks for the article post.Thanks Again. Much obliged.

15. website. Reading this information So i am glad to convey that I have a very excellent uncanny feeling

16. health says:

Looking forward to reading more. Great article post. Really Great.

17. UoN says:

metal detector used for sale WALSH | ENDORA

18. bbw wife says:

Informative and precise Its difficult to find informative and accurate info but here I found

19. Way cool! Some very valid points! I appreciate

20. Frank Fabian says:

provide whether post dated check or authorize the borrowed funds company to electronically debit the total amount from your bank checking account.

21. You have touched some fastidious factors here.

22. I?аАТаЂаll right away snatch your rss as I can at to find your email subscription link or newsletter service. Do you have any? Kindly let me understand in order that I may just subscribe. Thanks.

23. Really informative blog post.Much thanks again. Want more.

25. Thanks for all аАааБТour vаА аЂаluablаА аЂа laboаА аБТ on this ?аА аЂаbsite.

26. your blog posts. After all I all be subscribing for your feed and I am hoping you write again very soon!

27. You are my breathing in, I possess few web logs and rarely run out from to brand.

28. e cig says:

niche. Your blog provided us valuable information to work on.

29. pretty practical stuff, overall I feel this is worthy of a bookmark, thanks

30. A round of applause for your blog post.Thanks Again. Want more.

31. wonderful points altogether, you simply won a logo new reader. What might you recommend about your publish that you just made a few days in the past? Any certain?

32. love dolls says:

33. junk yard says:

on quite a few of your posts. Several of them are rife with

34. I appreciate you sharing this blog post.Really looking forward to read more. Much obliged.

35. Thank you for your post.Thanks Again. Really Cool.

36. archangel says:

Woah! I am really loving the template/theme of this site. It as simple, yet effective. A lot of times it as difficult to get that perfect balance between usability and appearance.

37. Wow, great article.Much thanks again. Cool.

38. I really like and appreciate your blog. Keep writing.

39. Just Browsing While I was surfing today I noticed a excellent post about

40. Thanks so much for the article post.Really looking forward to read more. Really Cool.

41. Very great post. I just stumbled upon your blog and wanted

42. Photography says:

Major thanks for the article.Much thanks again. Much obliged.

43. parents says:

A round of applause for your article post. Great.

44. This brief posting can guidance you way in oral treatment.

45. Regards for helping out, excellent info. аЂааЂ You must do the things you think you cannot do.аЂ аЂа by Eleanor Roosevelt.

46. Maintenance says:

I cannot thank you enough for the blog. Will read on…

47. Muchos Gracias for your blog post. Really Great.

48. Only a smiling visitor here to share the love (:, btw outstanding style and design.

49. It as hard to come by knowledgeable people in this particular subject, however, you seem like you know what you are talking about! Thanks

50. Wow! Thank you! I continually needed to write on my blog something like that. Can I include a part of your post to my site?

51. MICE says:

Thanks so much for the blog article.Thanks Again. Will read on

52. This particular blog is no doubt educating and besides amusing. I have found a bunch of interesting things out of this blog. I ad love to go back again soon. Cheers!

53. You completed a number of first rate points near. I appeared by the internet for the problem and found the majority folks will go along with along with your website.

54. Thanks for another great post. Where else may anybody get that type of info in such an ideal way of writing? I have a presentation next week, and I am at the search for such information.

55. You could certainly see your skills within the work you write. The world hopes for more passionate writers such as you who are not afraid to mention how they believe. At all times go after your heart.

57. Thanks again for the article.Thanks Again. Keep writing.

58. You made some first rate points there. I appeared on the internet for the problem and found most individuals will go along with along with your website.

59. love dolls says:

Usually I don at learn article on blogs, but I wish to say that this write-up very compelled me to take a look at and do it! Your writing taste has been surprised me. Thanks, very nice post.

60. junk my car says:

Sites we like the time to read or visit the content or sites we have linked to below the

61. vortex pipe says:

I truly appreciate this blog.Thanks Again. Really Cool.

62. This is exactly what I was searching for, many thanks

63. This unique blog is without a doubt entertaining and factual. I have picked many handy tips out of this source. I ad love to go back over and over again. Thanks a lot!

64. Nice post. I learn something totally new and challenging on blogs I stumbleupon every day. It as always exciting to read through content from other writers and practice something from their web sites.

65. It as hard to find knowledgeable individuals inside this topic, however you be understood as guess what occurs you are discussing! Thanks

you are in point of fact a good webmaster. The site loading speed is incredible.

67. Fidget toy says:

Way cool! Some very valid points! I appreciate you writing this post and the rest of the site is also very good.

68. Sinhce the admin of this site iss working, no hesitation very

69. Thanks for some other great post. Where else may anybody get that kind of information in such an ideal method of writing? I ave a presentation next week, and I am at the look for such information.

70. ski japan says:

71. I value the post.Really thank you! Fantastic.

72. Some genuinely interesting info , well written and generally user genial.

73. This is one awesome blog post.Really looking forward to read more. Much obliged.

74. Take a look at my website as well and let me know what you think.

75. Thanks so much for the blog post.Really looking forward to read more. Fantastic.

76. Very good information. Lucky me I ran across your site by chance (stumbleupon). I have book-marked it for later!

78. This is really interesting, You are a very skilled blogger. I have joined your rss feed and look forward to seeking more of your fantastic post. Also, I ave shared your website in my social networks!

79. that may be the finish of this post. Right here you

80. Very interesting information!Perfect just what I was searching for! Fear not that thy life shall come to an end, but rather fear that it shall never have a beginning. by John Henry Cardinal Newman.

81. Taxis Beddau says:

82. Well I sincerely liked reading it. This article offered by you is very useful for accurate planning.

83. wonderful put up, very informative. I wonder why the other specialists of this sector don at understand this. You should continue your writing. I am sure, you have a great readers a base already!

84. of writing? I have a presentation subsequent week,

85. pre-workouts says:

Wow! This could be one particular of the most beneficial blogs We have ever arrive across on this subject. Actually Fantastic. I am also a specialist in this topic so I can understand your effort.

86. hand spinner says:

Im grateful for the article.Much thanks again. Really Cool.

87. Well I truly liked studying it. This tip offered by you is very effective for correct planning.

88. Shiva habitait dans etait si enthousiaste, conducteur irait ranger de

89. Im grateful for the blog.Really looking forward to read more. Keep writing.

90. Just Browsing While I was surfing yesterday I noticed a excellent post about

Of course, what a fantastic website and revealing posts, I will bookmark your website.All the Best!

92. more details says:

pretty beneficial material, overall I believe this is worth a bookmark, thanks

93. I truly appreciate this post.Much thanks again. Great.

94. Muchos Gracias for your blog article.Really thank you! Awesome.

95. You have remarked very interesting details! ps nice site.

96. Wonderful blog! I saw it at Google and I must say that entries are well thought of. I will be coming back to see more posts soon.

98. This very blog is definitely entertaining additionally informative. I have picked a bunch of helpful tips out of this blog. I ad love to go back every once in a while. Cheers!

99. Incredible quest there. What occurred after? Take care!

100. wow, awesome blog article.Thanks Again. Great.

101. This post is invaluable. When can I find out more?

102. I cannot thank you enough for the blog post.Really looking forward to read more. Cool.

103. Wow, wonderful blog layout! How long have you been running a blog for? you make blogging glance easy. The entire glance of your web site is great, as well as the content!

104. Im grateful for the blog.Really thank you! Fantastic.

105. Nice blog here! Also your site loads up very fast! What host are you using? Can I get your affiliate link to your host? I wish my web site loaded up as fast as yours lol

106. I really enjoy the blog post.Thanks Again. Want more.

107. Nelida says:

iаЂа?Splendid post writing. I concur. Visit my blog for a free trial now! Enjoy secret enlargement tips. Get big and rich. Did I mention free trial? Visit now.

108. Way cool! Some extremely valid points! I appreciate you writing this post and the rest of the site is very good.

109. Thanks again for the blog post.Much thanks again. Really Great.

110. Gertrudis says:

Normally I do not learn article on blogs, however I wish to say that this write-up very compelled me to take a look at and do so! Your writing style has been amazed me. Thanks, quite great article.

111. Entry-level says:

Very good blog.Thanks Again. Cool.

112. Delila says:

Wow, fantastic blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your site is fantastic, let alone the content!

113. Im obliged for the blog article.Thanks Again. Great.

114. Im grateful for the blog.Thanks Again. Really Great.

115. Music says:

Some really prime posts on this site, saved to bookmarks.

116. You ought to take part in a contest for among the most effective blogs on the web. I will suggest this internet website!

117. This article has truly peaked my interest. I am going to take a note of your site and keep checking for new details about once per week. I opted in for your RSS feed too.

Great post.Much thanks again. Fantastic.

119. amiclubwear says:

Pretty! This was an incredibly wonderful post. Many thanks for supplying this info.

120. scar treatment for acne scar treatment lotion scar treatment

121. #vegas says:

Very neat post.Really looking forward to read more. Cool.

122. That is a very good tip particularly to those fresh to the blogosphere. Brief but very accurate information Many thanks for sharing this one. A must read article!

123. Wohh precisely what I was searching for, thankyou for putting up. Talent develops in tranquillity, character in the full current of human life. by Johann Wolfgang von Goethe.

124. Looking forward to reading more. Great article.Thanks Again. Will read on

125. Way cool! Some extremely valid points! I appreciate you writing this write-up plus the rest of the site is extremely good.

126. Very good article.Much thanks again. Fantastic.

127. I truly enjoаАааБТe? reading it, you could be a great author.

128. Wow, marvelous blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your website is great, as well as the content!. Thanks For Your article about sex.

129. Wow, fantastic weblog structure! How lengthy have you been running a blog for? you make blogging look easy. The entire glance of your site is great, let alone the content material!

130. Maybe you can write subsequent articles relating to this

132. I truly appreciate this post. I?аАТаЂаve been looking everywhere for this! Thank goodness I found it on Bing. You ave made my day! Thank you again

133. I value the post.Really looking forward to read more. Great.

134. This site truly has all of the information and facts I wanted about this subject and didn at know who to ask.

135. This is precisely what I used to be searching for, thanks

136. Regards for this post, I am a big big fan of this website would like to go on updated.

137. I think other site proprietors should take this website as an model, very clean and excellent user genial style and design, let alone the content. You are an expert in this topic!

138. joojav xxx says:

This excellent website certainly has all of the information and facts I wanted concerning this subject and didn at know who to ask.

139. flexvit says:

Thanks so much for the blog article.Really thank you! Cool.

140. Well I truly liked reading it. This tip provided by you is very useful for proper planning.

141. elektronik sigaralar says:

Very neat blog post.Thanks Again. Keep writing.

142. will go along with your views on this website.

143. It’аs really a nice and useful piece of information. I’аm satisfied that you shared this useful information with us. Please keep us up to date like this. Thank you for sharing.

144. You made some really good points there. I checked on the internet for additional information about the issue and found most individuals will go along with your views on this website.

145. Spokane HVAC says:

that as equally educative and engaging, and let

Thank you ever so for you article post. Fantastic.

149. Morbi molestie fermentum sem quis ultricies

150. This is a topic that is near to my heart Thank you!

151. The Silent Shard This may most likely be very handy for a few of your work opportunities I intend to you should not only with my blogging site but

152. Useful information for all Great remarkable issues here. I am very satisfied to look your article. Thanks a lot and i am taking a look ahead to touch you. Will you kindly drop me a e-mail?

153. Some genuinely good information, Gladiolus I noticed this.

154. soon it wilpl be well-known, due to itss feature contents.

155. you. This is really a tremendous web site.

156. WWW says:

regular basis. It includes good material.

157. Ia??a?аАааАТаЂ ve recently started a site, the information you provide on this site has helped me tremendously. Thanks for all of your time & work.

158. Thank you ever so for you article post.Much thanks again. Cool.

159. ebooki.net says:

I reckon something really interesting about your web blog so I saved to my bookmarks.

160. Wow! This could be one particular of the most beneficial blogs We have ever arrive across on this subject. Basically Excellent. I am also a specialist in this topic so I can understand your hard work.

161. Frp Grating says:

Well I sincerely liked studying it. This information procured by you is very effective for correct planning.

162. Scissor Lift says:

Major thankies for the article post.Thanks Again. Awesome.

163. Only wanna remark that you have a very nice site, I love the design and style it really stands out.

164. Thanks for sharing your thoughts. I really appreciate your efforts and I am waiting for your further post thanks once again.

165. Thanks for sharing this excellent post. Very interesting ideas! (as always, btw)

166. Maryjo says:

Wow, great blog post.Really looking forward to read more.

167. Many thanks for sharing this very good article. Very interesting ideas! (as always, btw)

168. indeed, investigation is having to pay off. So happy to possess found this article.. of course, analysis is having to pay off. Wonderful thoughts you possess here..

169. Very good information. Lucky me I discovered your site by chance (stumbleupon). I have bookmarked it for later!

170. Yahoo results While browsing Yahoo I discovered this page in the results and I didn at think it fit

171. junk my car says:

What as Taking place i am new to this, I stumbled upon this I ave found It absolutely useful and it has aided me out loads. I hope to contribute & assist other customers like its aided me. Good job.|

172. Thank you for the auspicious writeup. It in fact was a amusement account it. Look advanced to more added agreeable from you! By the way, how could we communicate?

173. Wonderful article! We will be linking to this great article on our site. Keep up the great writing.

174. tiffany rings Secure Document Storage Advantages | West Coast Archives

175. BMW repair says:

176. Its hard to find good help I am constantnly proclaiming that its hard to find good help, but here is

177. music says:

You made some nice points there. I looked on the internet for the subject matter and found most individuals will approve with your website.

178. Rattling good information can be found on weblog.

179. This awesome blog is really awesome and besides amusing. I have discovered helluva handy advices out of this amazing blog. I ad love to visit it every once in a while. Cheers!

180. It as laborious to seek out knowledgeable folks on this subject, however you sound like you recognize what you are speaking about! Thanks

181. they have been a moment to consider taking a shot?

182. whoah this weblog is great i love reading your posts. Stay

183. Wow, awesome blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your site is great, as well as the content!

184. Perfect piece of work you have done, this internet site is really cool with excellent information.

185. Wow, great article.Really looking forward to read more. Great.

186. That is a good tip particularly to those fresh to the blogosphere. Brief but very accurate info Thanks for sharing this one. A must read post!

187. black magic says:

188. Major thanks for the article.Really looking forward to read more. Want more.

189. Major thanks for the article post.Really thank you! Awesome.

190. Very neat post.Really thank you! Cool.

191. Enjoyed every bit of your blog article. Really Great.

192. Thanks-a-mundo for the article post.Really thank you! Much obliged.

193. Wow, great blog.Much thanks again. Much obliged.

194. Thanks for sharing, this is a fantastic post.Much thanks again. Awesome.

195. App web says:

Only a smiling visitant here to share the love (:, btw outstanding pattern. Make the most of your regrets. To regret deeply is to live afresh. by Henry David Thoreau.

196. Im grateful for the blog post.Thanks Again. Really Cool.

197. E-commerce says:

Im obliged for the blog post.Really thank you!

198. wow, awesome article post.Really thank you! Cool.

199. I really like and appreciate your post.Thanks Again. Really Great.

200. informatii interesante si utile postate pe blogul dumneavoastra. dar ca si o paranteza , ce parere aveti de inchirierea apartamente vacanta ?.

201. I think other site proprietors should take this site as an model, very clean and magnificent user friendly style and design, let alone the content. You are an expert in this topic!

Wow, great blog article.Really thank you! Really Great.

203. lot of work? I am brand new to blogging but I do write in my diary

204. Thank you for your post. Much obliged.

205. pretty valuable stuff, overall I consider this is worth a bookmark, thanks

206. Really informative article post.Thanks Again. Great.

207. Thanks for the post. I will definitely comeback.

208. junk my car says:

It as hard to discover knowledgeable folks on this subject, but you sound like you know what you are talking about! Thanks

209. Really informative post.Thanks Again. Will read on

210. I see something really interesting about your web blog so I saved to my bookmarks.

211. We stumbled over here different page and thought I might as well check things out. I like what I see so now i am following you. Look forward to exploring your web page repeatedly.

212. Well I found this on Digg, and I like it so I dugg it!

213. you can look here How do you password protect a Blogger blog on a custom domain?

214. Simply wanna remark that you have a very decent site, I the design it really stands out.

215. Audi repair says:

I relish, cause I discovered exactly what I was looking for. You have ended my four day long hunt! God Bless you man. Have a great day. Bye

216. Your means of describing the whole thing in this paragraph is really good, every one be able to simply know it, Thanks a lot.

217. Thanks for sharing, this is a fantastic blog post.Really looking forward to read more. Great.

218. tottenham hotspur jersey ??????30????????????????5??????????????? | ????????

219. You ave made some good points there. I looked on the web to learn more about the issue and found most individuals will go along with your views on this site.

220. This web site truly has all of the info I wanted concerning this subject and didn at know who to ask.

221. I really liked your blog.Much thanks again. Cool.

222. one and i was just wondering if you get a lot of spam responses?

223. My brother recommended I might like this web site. He was entirely right. This post truly made my day. You cann at imagine simply how much time I had spent for this info! Thanks!

224. Very nice write-up. I certainly love this website. Thanks!

226. this website says:

I value the blog post.Really thank you! Awesome.

227. I cannot thank you enough for the article post.Thanks Again. Want more.

228. Informative article, totally what I needed.

229. backpacking says:

Major thanks for the article.Really thank you! Much obliged.

230. Very good blog post. I definitely love this site. Stick with it!

231. This is one awesome article post. Really Great.

232. Way cool! Some very valid points! I appreciate you penning this post and also the rest of the site is also very good.

233. Thanks, I ave recently been searching for facts about this subject for ages and yours is the best I ave found so far.

234. little bit acquainted of this your broadcast provided bright clear idea