The Main Idea

Double-angle formulas let us evaluate trig functions of angles like [latex]2\theta[/latex] by rewriting them in terms of [latex]\sin\theta[/latex] and [latex]\cos\theta[/latex]. They are especially useful for finding exact values of angles not directly on the unit circle, such as [latex]\sin(2 \cdot 15^\circ)[/latex] or [latex]\cos(2 \cdot 22.5^\circ)[/latex].

The formulas are:

• [latex]\sin(2\theta) = 2\sin\theta\cos\theta[/latex]

• [latex]\cos(2\theta) = \cos^{2}\theta - \sin^{2}\theta = 2\cos^{2}\theta - 1 = 1 - 2\sin^{2}\theta[/latex]

• [latex]\tan(2\theta) = \dfrac{2\tan\theta}{1 - \tan^{2}\theta}[/latex]

Quick Tips: Using Double-Angle Formulas

1. Pick the Right Version of Cosine

   • Use [latex]2\cos^{2}\theta - 1[/latex] if cosine is easy to compute.

   • Use [latex]1 - 2\sin^{2}\theta[/latex] if sine is easier.

2. Work an Example with Special Angles

   • Find [latex]\sin(30^\circ)[/latex] using double-angle:

   • [latex]\sin(30^\circ) = \sin(2 \cdot 15^\circ)[/latex]
     = [latex]2\sin(15^\circ)\cos(15^\circ)[/latex].

   • Use known exact values of [latex]\sin(15^\circ)[/latex] and [latex]\cos(15^\circ)[/latex] to evaluate.

3. Simplify Step by Step

   • Always reduce radicals and fractions fully.

   • Keep answers exact in the form of fractions and square roots.

4. Check the Quadrant

   • The sign of [latex]\sin(2\theta)[/latex], [latex]\cos(2\theta)[/latex], or [latex]\tan(2\theta)[/latex] depends on which quadrant [latex]2\theta[/latex] is in.

5. Example: Find [latex]\cos(2 \cdot 22.5^\circ)[/latex].

   • [latex]\cos(45^\circ) = \cos^{2}(22.5^\circ) - \sin^{2}(22.5^\circ)[/latex].

   • Since [latex]\cos(45^\circ) = \dfrac{\sqrt{2}}{2}[/latex], this connects the exact values of half-angles to double-angles.

The Main Idea

Double-angle formulas connect trigonometric functions of [latex]2\theta[/latex] to those of [latex]\theta[/latex]. They are powerful tools for proving that two trig expressions are equal. In verification problems, you expand one side using a double-angle identity, then simplify until it matches the other side. This shows that the identity holds for all values where both sides are defined.

The formulas are:

• [latex]\sin(2\theta) = 2\sin\theta\cos\theta[/latex]

• [latex]\cos(2\theta) = \cos^{2}\theta - \sin^{2}\theta = 2\cos^{2}\theta - 1 = 1 - 2\sin^{2}\theta[/latex]

• [latex]\tan(2\theta) = \dfrac{2\tan\theta}{1 - \tan^{2}\theta}[/latex]

Quick Tips: Using Double-Angle Formulas in Verifications

1. Choose the Right Identity

   • If the expression involves [latex]\sin\theta\cos\theta[/latex], use [latex]\sin(2\theta)[/latex].

   • If it involves [latex]\cos^{2}\theta[/latex] or [latex]\sin^{2}\theta[/latex], use one of the cosine formulas.

   • If tangent appears, use [latex]\tan(2\theta)[/latex].

2. Work From the Complicated Side

   • Expand or replace terms on the more complex side first.

   • Leave the simpler side as the “target.”

3. Simplify Step by Step

   • Apply Pythagorean identities to replace [latex]\sin^{2}\theta[/latex] or [latex]\cos^{2}\theta[/latex] if needed.

   • Combine fractions or cancel factors carefully.

The Main Idea

Reduction formulas (also called power-reducing formulas) allow us to rewrite even powers of sine or cosine in terms of the first power of cosine. These formulas are derived from the double-angle formulas and are especially important in calculus.

The reduction formulas are:

• [latex]\sin^2\theta = \frac{1 - \cos(2\theta)}{2}[/latex]
• [latex]\cos^2\theta = \frac{1 + \cos(2\theta)}{2}[/latex]
• [latex]\tan^2\theta = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}[/latex]

Quick Tips: Using Reduction Formulas

• Apply the formula to reduce the power
• If higher powers remain, apply the formula again
• Simplify using algebra

The Main Idea

Half-angle formulas allow us to evaluate trig functions of angles like [latex]\dfrac{\theta}{2}[/latex] by rewriting them in terms of [latex]\sin\theta[/latex] and [latex]\cos\theta[/latex]. They are especially useful for finding exact values of angles not directly on the unit circle, such as [latex]22.5^\circ[/latex] or [latex]\dfrac{\pi}{8}[/latex].

The formulas are:

• [latex]\sin\left(\dfrac{\theta}{2}\right) = \pm\sqrt{\dfrac{1-\cos\theta}{2}}[/latex]

• [latex]\cos\left(\dfrac{\theta}{2}\right) = \pm\sqrt{\dfrac{1+\cos\theta}{2}}[/latex]

• [latex]\tan\left(\dfrac{\theta}{2}\right) = \dfrac{\sin\theta}{1+\cos\theta} = \dfrac{1-\cos\theta}{\sin\theta}[/latex]

The sign (positive or negative) depends on the quadrant of [latex]\dfrac{\theta}{2}[/latex].

Quick Tips: Using Half-Angle Formulas

1. Check the Quadrant

   • Decide whether [latex]\dfrac{\theta}{2}[/latex] is in Quadrant I, II, III, or IV.

   • Use this to assign the correct sign in the formula.

2. Choose the Formula That Fits Best

   • For sine and cosine, use the square root forms.

   • For tangent, pick the version that avoids dividing by zero.