Exercises


  1. Use an appropriate double angle formula to rewrite each expression as a single trigonometric ratio.
    1. \( 6 \sin{(3x)} \cos{(3x)} \)
    2. \( 1-2 \cos^2{\left(\frac{\alpha}{2}\right)} \)
    3. \( \dfrac{\tan{(\theta)}}{\tan^2{(\theta)}-1} \)
    4. \( \dfrac{\cos^2{(3y)}-\sin^2{(3y)}}{\sin{(3y)}\cos{(3y)}} \)
  2. Simplify each of the following trigonometric expressions using an appropriate double angle formula, then determine the exact value of the expression.
    1. \( 1-2 \sin^2{\left(\dfrac{11\pi}{8}\right)} \)
    2. \( 1 - 2\cos^2{(105^\circ)} \)
    3. \( 4 \sin (112.5^\circ) \cos (112.5^\circ) \)
    4. \( \dfrac{1-\tan^2{\left(\frac{\pi}{12}\right)}}{\tan{\left(\frac{\pi}{12}\right)}} \)
    5. \( \cos^2(\dfrac{11\pi}{12}) \)
  3. Rewrite each of the following trigonometric ratios using a double angle formula for sine, cosine or tangent.
    1. \( \cos{(4A)} \)
    2. \( \sin {(3\alpha)} \)
    3. \( \cos{(-x)} \)
    4. \( \cot {\left(\dfrac{\theta}{2}\right)} \)
    1. Given \( \sin{(\theta)}=\dfrac{3}{4} \) where \( \frac{\pi}{2} \leq \theta \leq \pi \), determine the exact value of \( \sin{(2\theta)} \).
    2. Given \( \cos{(2\theta)}=-\dfrac{7}{8} \), where \( 2\theta \) is an angle in standard position with a terminal arm in quadrant 3, determine the exact value of \( \cos{(\theta)} \) and \( \sin{(\theta)} \).
  5. Given \( \sin{(100^\circ)}=k \), determine an expression for each of the following in terms of \( k \).
    1. \( \cos{(200^\circ)} \)
    2. \( \sin{(200^\circ)} \)
    3. \( \sin{(20^\circ)} \)
  6. Determine the exact value of each.
    1. \( \cos{(22.5^\circ)} \)
    2. \( \sin {\left(\dfrac{7\pi}{12}\right)} \)
    3. \( \tan{\left(\dfrac{5\pi}{8}\right)} \)
  7. Prove the following.
    1. \( \cos^4{(\beta)}-\sin^4{(\beta)}=\cos{(2\beta)} \)
    2. \( \cos{(2\theta)} = \dfrac{1-\tan^2{(\theta)}}{1+\tan^2{(\theta)}} \)
    3. \( \tan (x) + \cot (x) = 2\csc(2x) \)
    4. \( \tan (y) = \dfrac{\sin (2y) + \sin (y)}{1+\cos (y) + \cos (2y)} \)
    5. \( \dfrac{2\cos {(2\theta)}}{\sin{(2\theta)} - 2 \sin^2{(\theta)}}=1+\cot{(\theta)} \)
    6. \( \sec{(2x)} = \dfrac{ \csc^2 {(x)}}{\csc^2{(x)}-2} \)
    7. \( 2\sin{(\theta+\beta)}\cos{(\theta-\beta)}=\sin{(2\theta)}+\sin{(2\beta)} \)
    8. \( \dfrac{1+\sin{(2x)}-\cos{(2x)}}{1+\sin{(2x)}+\cos{(2x)}}=\tan{(x)} \)
  8. Derive a formula for
    1. \( \sin{(3\theta)} \) in terms of \( \sin{(\theta)} \)
    2. \( \cos{(4\theta)} \) in terms of \( \cos{(\theta)} \)
    3. \( \cot{(2\theta)} \) in terms of \( \cot{(\theta)} \)
  9. Right triangle with side lengths a, b, c, angle C is right angle For a right triangle \( ABC \) (shown to the right) with \( \angle C = \frac{\pi}{2} \), show that the area of the triangle is given by \( \dfrac{1}{4}c^2\sin{(2A)} \) or \( \dfrac{1}{4}c^2\sin{(2B)} \).
     
  10. Using appropriate double angle formulas and your knowledge of transformations, sketch the graphs of \( f(x)= 2\sin^2{(x)} \) and \( g(x)=3\sin{(2x)}\cos{(2x)}-1 \).