Trigonometric Functions

1.  If \( \displaystyle{P(x,y) = \left(-\frac{1}{2}, \frac{\sqrt 3}{2} \right)} \) is the point on the unit circle that corresponds to a real number \(t\). Find the exact values of the six trigonometric functions of \(t\).   SOLUTION

2. Given that \(\sin \theta = \frac{3}{5}\), and \(\theta\) is an acute angle, find the exact value of each of the remaining five trigonometric functions of \(\theta\).   SOLUTION

3. Find the exact value of each of the remaining trigonometric functions of \(\theta\) if \(\displaystyle{\sin\theta = \frac{12}{13}} \) and \(\displaystyle{\tan \theta < 0 } \).   SOLUTION

4. Find the exact values of the six trigonometric functions of an angle \(\theta \)  if \((-4,3) \) is a point on its terminal side in standard position.   SOLUTION

5. Find the exact value of each expression. Do not use a calculator.  SOLUTION

(a)  \(\displaystyle{\tan 10^\circ -\frac{\sin 10^\circ}{\cos 10^\circ}}\)    (b) \(\displaystyle{\cos^2 \frac{\pi}{5} +\frac{1}{\csc^2 \frac{\pi}{5}}}\)    (c)  \(\displaystyle{\sec^2 40^\circ - \tan^2 40^\circ}\)     (d)  \(\displaystyle{\tan 10^\circ \cos 10^\circ \csc 10^\circ}\)

6. The builder of a parking garage wants to build a ramp that meets the ground at \(30^\circ\). If the horizontal span covers \(30\) feet, how long is the ramp?  SOLUTION

7. Answer the following.   SOLUTION

(a) Given that \(\cos\theta=\dfrac{2}{3}\). Find the exact value of \( \cos\theta + \cos(\theta+2\pi) + \cos(\theta+4\pi)\quad \)

(b) Given that \(\cot\theta= -2\). Find the exact value of \( \cot\theta + \cot(\theta-\pi) + \cot(\theta-2\pi)\quad \)