📏Honors Pre-Calculus Unit 7 Review

Double-angle formulas let you rewrite trig functions of $2\theta$ using only trig functions of $\theta$ . These come up constantly when simplifying expressions and verifying identities.

Here are the three core formulas:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$ $cos (2 θ) = cos^{2} (θ) - sin^{2} (θ)$
- Also written as $\cos(2\theta) = 2\cos^2(\theta) - 1$ or $\cos(2\theta) = 1 - 2\sin^2(\theta)$
- The alternate forms are derived using the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ . Choose whichever version matches what you already know (just sine, just cosine, or both).
$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

To calculate an exact value using a double-angle formula:

Identify the angle $\theta$ so that the target angle equals $2\theta$ . For example, if you need $\sin(120°)$ , set $\theta = 60°$ .
Substitute the known values of $\sin(\theta)$ , $\cos(\theta)$ , or $\tan(\theta)$ into the appropriate formula.
Simplify the expression to get your exact answer.

To verify an identity using double-angle formulas:

Replace any double-angle expressions (like $\sin(2\theta)$ ) with their equivalent single-angle forms.
Simplify each side of the identity independently.
Show that both sides reduce to the same expression.

Example: Verify that $\sin(2\theta) = \frac{2\tan(\theta)}{1 + \tan^2(\theta)}$ .

Start with the right side. Rewrite $\tan(\theta)$ as $\frac{\sin(\theta)}{\cos(\theta)}$ :

$\frac{2 \cdot \frac{\sin(\theta)}{\cos(\theta)}}{1 + \frac{\sin^2(\theta)}{\cos^2(\theta)}} = \frac{\frac{2\sin(\theta)}{\cos(\theta)}}{\frac{\cos^2(\theta) + \sin^2(\theta)}{\cos^2(\theta)}} = \frac{2\sin(\theta)}{\cos(\theta)} \cdot \frac{\cos^2(\theta)}{1} = 2\sin(\theta)\cos(\theta) = \sin(2\theta)$

Half-Angle Formulas

Utilize half-angle formulas to determine precise trigonometric values in complex expressions

Half-angle formulas express trig functions of $\frac{\theta}{2}$ in terms of trig functions of $\theta$ . They're especially useful for finding exact values of angles like $15°$ or $22.5°$ that aren't on the standard unit circle.

$\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}$
$\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$
$\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$ $tan (\frac{θ}{2}) = \pm \frac{1 - c o s ( θ )}{1 + c o s ( θ )}$
- Two alternate forms that avoid the $\pm$ : $\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}$

The $\pm$ in front of the square root is not optional decoration. You must choose the correct sign based on which quadrant $\frac{\theta}{2}$ falls in.

To find an exact value using a half-angle formula:

Identify the original angle $\theta$ so that your target angle equals $\frac{\theta}{2}$ . For example, to find $\cos(15°)$ , set $\theta = 30°$ .
Substitute the known value of $\cos(\theta)$ (or $\sin(\theta)$ ) into the appropriate formula.
Simplify the expression under the square root.
Determine the sign: figure out which quadrant $\frac{\theta}{2}$ lies in, then apply the correct sign for that trig function in that quadrant.

Example: Find the exact value of $\sin(105°)$ .

Set $\theta = 210°$ , so $\frac{\theta}{2} = 105°$ . Since $105°$ is in Quadrant II, sine is positive.

$\sin(105°) = +\sqrt{\frac{1 - \cos(210°)}{2}} = \sqrt{\frac{1 - (-\frac{\sqrt{3}}{2})}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2}$

Reduction Formulas

Implement reduction formulas to simplify trigonometric expressions and solve equations

Reduction formulas (also called cofunction identities) let you convert between complementary trig functions. They're based on the fact that cofunctions of complementary angles are equal.

$\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right)$
$\cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right)$
$\tan(\theta) = \cot\left(\frac{\pi}{2} - \theta\right)$
$\csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right)$
$\sec(\theta) = \csc\left(\frac{\pi}{2} - \theta\right)$
$\cot(\theta) = \tan\left(\frac{\pi}{2} - \theta\right)$

Notice the pattern: each function equals its cofunction evaluated at the complementary angle. Sine pairs with cosine, tangent pairs with cotangent, secant pairs with cosecant.

To simplify using reduction formulas:

Identify the trig function and angle in your expression.
Choose the reduction formula that converts it to a more useful form.
Substitute and simplify using known values or other identities.

For solving equations: Apply reduction formulas to rewrite the equation so all terms use the same trig function, then isolate the variable and solve algebraically.

Use radian measure when working with these formulas, since it keeps calculations with periodic functions cleaner.

Applications and Comparisons

Apply double-angle formulas to calculate exact trigonometric values and verify identities, Trigonometric functions - Wikipedia

Compare and contrast the applications of angle formulas vs reduction formulas

Each type of formula has a distinct purpose, and recognizing which one to use is half the battle.

Double-angle formulas are your go-to when you see $2\theta$ inside a trig function, or when you need to convert a product like $2\sin(\theta)\cos(\theta)$ into a single function. They also appear frequently in identity verification problems.
Half-angle formulas are most useful when you need exact values for angles that are half of a known angle (like finding $\sin(22.5°)$ from $\cos(45°)$ ), or when simplifying expressions that contain $\frac{\theta}{2}$ .
Reduction formulas help when you need to rewrite a trig function as its cofunction, which is particularly handy for combining terms in an equation or simplifying expressions with angles outside the first quadrant.

Many problems require combining these tools. You might use a double-angle formula to expand an expression, then apply a reduction formula to simplify the result further. The key is to look at the structure of the problem: Does it involve $2\theta$ ? Use double-angle. Does it involve $\frac{\theta}{2}$ ? Use half-angle. Do you need to switch between cofunctions? Use reduction formulas.

Fundamental Concepts

Understand the relationship between trigonometric functions and the unit circle

The unit circle is the foundation for everything in this section. Each point on the unit circle corresponds to $(\cos(\theta), \sin(\theta))$ , which is where all these formulas ultimately come from.

Periodicity: Sine and cosine repeat every $2\pi$ radians ( $360°$ ). This periodic behavior is what makes reduction formulas work, since angles that differ by full periods produce the same trig values.
Even and odd properties: Cosine is an even function, meaning $\cos(-\theta) = \cos(\theta)$ . Sine and tangent are odd functions, meaning $\sin(-\theta) = -\sin(\theta)$ and $\tan(-\theta) = -\tan(\theta)$ . These properties help you determine signs when using half-angle formulas or working with negative angles.

📏Honors Pre-Calculus Unit 7 Review