Trigonometry Questions & Answers

4. Given that \( \sin x=\frac{3}{5} \) and that \( 0<x<\frac{\pi}{2} \), find the exact values of: \( \begin{array}{lll}\text { (a) } \cos x & \text { (b) } \cos 2 x & \text { (c) } \sin 2 x\end{array} \)

3. Prove each co-function identity using the compound angle identities. (a) \( \tan \left(\frac{\pi}{2}-\theta\right)=\cot \theta \) (b) \( \sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta \) (c) \( \csc \left(\frac{\pi}{2}-\theta\right)=\sec \theta \) 4. Given that \( \sin x=\frac{3}{5} \) and that \( 0<x<\frac{\pi}{2} \), find the exact values of:

(1) Halla, en cada caso, el valor de las funciones trigo- nométricas a partir de \( P(x, y) \). \( \begin{array}{ll}\text { a. }\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right) & \text { b. }\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right) \\ \text { c. }\left(\frac{2}{3},-\frac{\sqrt{5}}{3}\right) & \text { d. }\left(\frac{3}{5}, \frac{4}{5}\right)\end{array} \)

2. Sketch a standard position angle with each radian measure. \( \begin{array}{lll}\text { a. } \frac{5 \pi}{4} & \text { b. }-\pi \\ \text { c. } \frac{-4 \pi}{3} & \text { d. } \frac{15 \pi}{4}\end{array} \) Convert to degrees, minutes, and seconds (DMS). \( \begin{array}{lll}\text { 3. }-2.87^{\circ} & \text { 4. } 110.51^{\circ} & \text { 5. } 48.362^{\circ}\end{array} \) Convert to decimal degrees (DD). \( \begin{array}{lll}\text { 6. }-58^{\circ} 54^{\prime} & \text { 7. } 98^{\circ} 45^{\prime} 43.2^{\prime \prime} & \text { 8. } 135^{\circ} 35^{\prime} 38.4^{\prime \prime}\end{array} \)

En los ejercicios 1 a 5 encuentre las soluciones de la ecuación en el intervalo \( 0^{\circ} \leq \theta \leq 360^{\circ} \) 1. \( \operatorname{sen} \theta=\frac{\sqrt{3}}{2} \) 2. \( \cos \theta-1=0 \) 3. \( \tan \theta+\sqrt{3}=0 \) 4. \( \sec \theta-\sqrt{2}=0 \) 5. \( \cot \theta=-1 \)

\( \begin{array}{ll}\text { IDENTIDADES TRIGONOMÉTRICAS } & \text { 8. }(\operatorname{sen} v+\cos v)^{2}+(\operatorname{sen} v-\cos v)^{2}=2 \\ \text { Demostrar las siguientes identidades } & \text { 9. }(\operatorname{sen} v+\operatorname{cosec} v)^{2}=\operatorname{sen}^{2} v+\operatorname{cotan}^{2} v+3 \\ \text { 1. } \cos v \tan v=\operatorname{sen} v & \text { 10. } \sec ^{2} v-\sec ^{2} v=\tan ^{4} v-\tan ^{2} v \\ \text { 2. } \cot v \cdot \tan v=1 & \text { 11. }(\sec v+\cos v)(\sec v-\cos v)=\tan ^{2} v+\operatorname{sen}^{2} v \\ \text { 3. } \operatorname{sen} v \cot v+\cos v \tan v=\operatorname{sen} v+\cos v & \text { 12. }\left(1+\tan ^{2} v\right) \cos ^{2} v=1 \\ \text { 4. } \operatorname{sen} v \sec v=\tan v & \text { 13. } \operatorname{sen}^{2} v+\operatorname{sen}^{2} v \tan ^{2} v=\tan ^{2} v\end{array} \)

14. \( \tan v+\operatorname{cotan} v=\sec v \operatorname{cosec} v \) 15. \( \operatorname{sen}^{2} v \cos ^{2} v+\cos ^{4} v=\cos ^{2} v \)

\( \begin{aligned} \text { a) } \quad \sin x & =0 \\ 0 & \leqslant x \\ 0 & \leqslant x \\ 0 & \leqslant 360^{\circ}\end{aligned} \)

5. \( \operatorname{sen} v \operatorname{cotan} v=\cos v \) 6. \( \operatorname{sen} v \tan v+\cos v=\sec v \) 7. \( \operatorname{csec} v-\operatorname{sen} v=\operatorname{cotan} v \cos v \) 8. \( (\operatorname{sen} v+\cos v)^{2}+(\operatorname{sen} v-\cos v)^{2}=2 \) 9. \( (\operatorname{sen} v+\operatorname{cosec} v)^{2}=\operatorname{sen}^{2} v+\operatorname{cotan}^{2} v+3 \) 10. \( \sec ^{4} v-\sec ^{2} v=\tan ^{4} v-\tan ^{2} v \) 11. \( \left(\sec ^{2} v+\cos ^{2} v\right)\left(\sec v-\cos ^{2} v\right)=\tan ^{2} v+\operatorname{sen}^{2} v \) 12. \( \left(1+\tan ^{2} v\right) \cos ^{2} v=1 \)

Sin utilizar tablas ni calculadora, calcula y simplifica: a) \( 3 \operatorname{sen} 30^{\circ}+6 \cos 45^{\circ} \) b) \( \operatorname{sen}^{2} 45^{\circ}+\cos ^{2} 45^{\circ} \) c) \( \tan ^{2} 60^{\circ}-\sec ^{2} 60^{\circ} \) d) \( \frac{\cos 60^{\circ}+\cos 30^{\circ}}{\csc ^{2} 30^{\circ}+\operatorname{sen}^{2} 45^{\circ}} \)