Study Guide: Double-Angle and Half-Angle Formulas in Precalculus

Q1. Find the exact value of each of the following. If $\tan \theta = \frac{3}{2}$ and $\frac{\pi}{6} < \theta < \frac{\pi}{2}$:

Background

Topic: Double-Angle Formulas

This question tests your ability to use double-angle identities for sine, cosine, and tangent to find exact values given a trigonometric ratio and a range for $\theta$.

Key formulas:

• $\sin(2\theta) = 2\sin\theta\cos\theta$

• $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

• $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Step-by-Step Guidance

1. Start by using the given $\tan\theta = \frac{3}{2}$ to find $\sin\theta$ and $\cos\theta$. Recall $\tan\theta = \frac{\sin\theta}{\cos\theta}$.

2. Construct a right triangle with opposite side 3, adjacent side 2, and hypotenuse $\sqrt{3^2 + 2^2} = \sqrt{13}$ to find $\sin\theta = \frac{3}{\sqrt{13}}$ and $\cos\theta = \frac{2}{\sqrt{13}}$.

3. Plug these values into the double-angle formulas for $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$.

4. For $\sin(2\theta)$: $\sin(2\theta) = 2 \cdot \frac{3}{\sqrt{13}} \cdot \frac{2}{\sqrt{13}}$.

5. For $\cos(2\theta)$: $\cos(2\theta) = \left(\frac{2}{\sqrt{13}}\right)^2 - \left(\frac{3}{\sqrt{13}}\right)^2$.

6. For $\tan(2\theta)$: $\tan(2\theta) = \frac{2 \cdot \frac{3}{2}}{1 - \left(\frac{3}{2}\right)^2}$.

Try solving on your own before revealing the answer!

Final Answers:

$\sin(2\theta) = \frac{12}{13}$

$\cos(2\theta) = \frac{-5}{13}$

$\tan(2\theta) = -\frac{12}{5}$

These values are found by plugging the triangle ratios into the double-angle formulas and simplifying.

Q2. Use the half-angle formulas to find the exact value of the trigonometric function.

Background

Topic: Half-Angle Formulas

This question tests your ability to use half-angle identities to find exact values for sine, cosine, and tangent of half-angles.

Key formulas:

• $\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}}$

• $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}$

• $\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$

Step-by-Step Guidance

1. Identify the angle for which you need to use the half-angle formula (e.g., $22.5^\circ$ or $\frac{5\pi}{12}$).

2. Find $\cos\theta$ for the original angle (e.g., $\cos(45^\circ)$ if $\theta = 45^\circ$).

3. Plug the value of $\cos\theta$ into the appropriate half-angle formula.

4. Choose the correct sign based on the quadrant in which the half-angle lies.

Try solving on your own before revealing the answer!

Final Answers:

$\sin(22.5^\circ) = \frac{\sqrt{2 - \sqrt{2}}}{2}$

$\tan\left(\frac{\pi}{8}\right) = \sqrt{2} - 1$

$\sin\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}$

These values are found by plugging the known cosine values into the half-angle formulas and simplifying.

Q3. Use the figure to evaluate each function.

Background

Topic: Trigonometric Functions and Double-Angle Identities

This question tests your ability to use geometric information (from a figure) to evaluate trigonometric functions, especially using double-angle identities.

Key formulas:

• $\sin(2\theta) = 2\sin\theta\cos\theta$

• $\cos(2\theta) = \cos^2\theta - \sin^2\theta$

Step-by-Step Guidance

1. Use the coordinates given in the figure to find $\sin\theta$ and $\cos\theta$ (e.g., $y$-coordinate over radius for $\sin\theta$).

2. Plug these values into the double-angle formulas to find $\sin(2\theta)$ and $\cos(2\theta)$.

3. Simplify the expressions using the values from the figure.

Try solving on your own before revealing the answer!

Final Answers:

$\sin(2\theta) = \frac{2a\sqrt{3}}{7}$

$\cos(2\theta) = \frac{a^2 - 3}{7}$

These values are found by plugging the coordinates into the double-angle formulas.

Q4. Establish the identity.

Background

Topic: Trigonometric Identities

This question tests your ability to manipulate and prove trigonometric identities using double-angle and reciprocal identities.

Key formulas:

• $\sec\theta = \frac{1}{\cos\theta}$

• $\csc\theta = \frac{1}{\sin\theta}$

• $\sec(2\theta)$ and $\csc(2\theta)$ can be written in terms of $\cos(2\theta)$ and $\sin(2\theta)$

Step-by-Step Guidance

1. Rewrite $\sec(2\theta)$ and $\csc(2\theta)$ in terms of $\cos(2\theta)$ and $\sin(2\theta)$.

2. Express $\cos(2\theta)$ and $\sin(2\theta)$ using double-angle formulas.

3. Manipulate the expressions to show the identity as required.

Try solving on your own before revealing the answer!

Final Answer:

The identity is established by expressing all terms in terms of $\sin\theta$ and $\cos\theta$ and simplifying.

Q5. Solve the equation on the interval $0 \leq \theta < 2\pi$.

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve equations involving double-angle identities and to find solutions within a specified interval.

Key formulas:

• $\cos(2\theta) = 2\cos^2\theta - 1$

• $\sin(2\theta) = 2\sin\theta\cos\theta$

Step-by-Step Guidance

1. Rewrite the equation using double-angle identities to express all terms in terms of $\sin\theta$ or $\cos\theta$.

2. Set the equation equal to zero and factor or use algebraic methods to solve for $\theta$.

3. Check which solutions are within the interval $0 \leq \theta < 2\pi$.

Try solving on your own before revealing the answer!

Final Answer:

Solutions are found by factoring and checking the interval. Answers are given in radians.

Q6. Find the exact value.

Background

Topic: Trigonometric Expressions and Double-Angle Identities

This question tests your ability to simplify trigonometric expressions using double-angle and half-angle formulas.

Key formulas:

• $\sin(2\theta) = 2\sin\theta\cos\theta$

• $\cos(2\theta) = 2\cos^2\theta - 1$

Step-by-Step Guidance

1. Identify the trigonometric expression and rewrite it using double-angle or half-angle formulas.

2. Simplify the expression step by step, using algebraic manipulation and known values.

Try solving on your own before revealing the answer!

Final Answer:

The exact value is found by substituting and simplifying using the formulas.

Q7. Rain gutter area maximization problem.

Background

Topic: Applications of Trigonometric Identities and Optimization

This question tests your ability to apply trigonometric identities to a real-world optimization problem, and to use calculus and graphing to find maximum values.

Key formulas:

• $A(\theta) = 25\sin\theta(\cos\theta + 1)$

• Equation to maximize: $\cos(2\theta) + \cos\theta = 0$

Step-by-Step Guidance

1. Set up the equation for area $A(\theta)$ and identify the interval for $\theta$.

2. Use double-angle identities to rewrite $\cos(2\theta)$ in terms of $\cos\theta$.

3. Solve the resulting equation for $\theta$ to find the angle that maximizes the area.

4. Plug the value of $\theta$ back into $A(\theta)$ to find the maximum area.

5. Use the graph to confirm the maximum and the corresponding angle.

Try solving on your own before revealing the answer!

Final Answer:

The maximum area and the angle are found by solving the trigonometric equation and checking the graph.

Q8. Find the value of the number C.

Background

Topic: Integrating Trigonometric Functions

This question tests your ability to compare two expressions involving trigonometric integrals and solve for an unknown constant.

Key formulas:

• Use the double-angle formula for $\cos(2x)$.

• Compare coefficients to solve for $C$.

Step-by-Step Guidance

1. Expand $\cos(2x)$ using the double-angle formula.

2. Compare the left and right sides of the equation to solve for $C$.

Try solving on your own before revealing the answer!

Final Answer:

$C = \frac{1}{6}$

This is found by matching coefficients after expanding the double-angle formula.