Trigonometry Solutions
Topics || Problems
With the use on different trigonometric identities, solutions to mathematical problems (with trigonometric problems) can be simplified.
I. BASIC TRIGONOMETRIC IDENTITIES
1. \(\sin \theta = \frac{1}{{\csc \theta }}\)
2. \(\cos \theta = \frac{1}{{\sec \theta }}\)
3. \(\tan \theta = \frac{1}{{\cot \theta }} = \frac{{\sin \theta }}{{\cos \theta }}\)
4. \(\cot \theta = \frac{1}{{\tan \theta }} = \frac{{\cos \theta }}{{\sin \theta }}\)
5. \({\sin ^2}\theta + {\cos ^2}\theta = 1\) Phytagorean identity
6. \(1 + {\tan ^2}\theta = {\sec ^2}\theta \)
7. \(1 + {\cot ^2}\theta = \csc \theta \)
II. SUM AND DIFFERENCE OF TWO ANGLES
8. \(Cos(A + B) = Cos\left( A \right)Cos\left( B \right) - Sin\left( A \right)Sin\left( B \right)\)
9. \(Cos(A - B) = Cos\left( A \right)Cos\left( B \right) + Sin\left( A \right)Sin\left( B \right)\)
10. \(Sin\left( {A + B} \right) = Sin\left( A \right)Cos\left( B \right) + Cos\left( A \right)Sin\left( B \right)\)
11. \(Sin\left( {A - B} \right) = Sin\left( A \right)Cos\left( B \right) - Cos\left( A \right)Sin\left( B \right)\)
12. \(Tan(A + B) = \frac{{Tan\left( A \right) + Tan\left( B \right)}}{{1 - Tan\left( A \right)Tan\left( B \right)}}\)
13. \(Tan(A - B) = \frac{{Tan\left( A \right) - Tan\left( B \right)}}{{1 + Tan\left( A \right)Tan\left( B \right)}}\)
III. DOUBLE ANGLE FORMULA
14. \(Cos\left( {2\theta } \right) = Co{s^2}\theta - Si{n^2}\theta \)
15. \(Cos\left( {2\theta } \right) = 2Co{s^2}\theta - 1\)
16. \(cos\left( {2\theta } \right) = 1 - 2{\sin ^2}\theta \)
17. \(Sin\left( {2\theta } \right) = 2Sin\theta Cos\theta \)
18. \(Tan\left( {2\theta } \right) = \frac{{2Tan\theta }}{{1 - Ta{n^2}\theta }}\)
IV. HALF ANGLE FORMULA
19. \(Sin\left( {\frac{\theta }{2}} \right) = \sqrt {\frac{{1 - Cos\theta }}{2}} \)
20. \(Cos\left( {\frac{\theta }{2}} \right) = \sqrt {\frac{{1 + Cos\theta }}{2}} \)
21. \(Tan\left( {\frac{\theta }{2}} \right) = \sqrt {\frac{{1 - Cos\theta }}{{1 + Cos\theta }}} \)
V. COS AND SIN LAW
22. Cosine Law
\({a^2} = {b^2} + {c^2} - 2bc\left[ {Cos\left( A \right)} \right]\)
\({b^2} = {a^2} + {c^2} - 2ac\left[ {Cos\left( B \right)} \right]\)
\({c^2} = {b^2} + {a^2} - 2ab\left[ {Cos\left( C \right)} \right]\)
23. Sine Law
\(\frac{a}{{\sin A}} = \frac{b}{{SinB}} = \frac{c}{{{\mathop{\rm sinC}\nolimits} }} \)
