Trigonometric Formulas Study Guide

Exam Tip: When working with the Unit Circle, remember that the $x$-coordinate corresponds to $\cos\theta$ and the $y$-coordinate corresponds to $\sin\theta$.

Trigonometric Identities

As you move into more advanced algebraic manipulation, you will need to use identities to simplify complex expressions. These are categorized into three main groups:

Reciprocal and Quotient Identities

Reciprocal identities allow you to rewrite functions in terms of their "flips":

• $\csc\theta = \frac{1}{\sin\theta}$
• $\sec\theta = \frac{1}{\cos\theta}$
• $\cot\theta = \frac{1}{\tan\theta}$

Quotient identities express tangent and cotangent using sine and cosine:

• $\tan\theta = \frac{\sin\theta}{\cos\theta}$
• $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Pythagorean Identities

These are perhaps the most critical identities for the midterm. They are derived from the Pythagorean Theorem applied to the unit circle:

1. $\sin^2\theta + \cos^2\theta = 1$
2. $1 + \cot^2\theta = \csc^2\theta$
3. $\tan^2\theta + 1 = \sec^2\theta$

Circular Functions and Arc Measurements

When we transition from triangles to circles, we often deal with "sectors" (slices of a circle) and "arcs" (the curved edge of a slice). A crucial rule for these calculations is that the angle $\theta$ must be in radians, not degrees.

Arclength and Sector Area

To find the length of an arc ($s$) given a radius ($r$) and an angle in radians ($\theta_{rad}$), use: $s = r \cdot \theta_{rad}$

To find the area ($A$) of a circular sector, use: $A = \frac{1}{2}r^2\theta_{rad}$

Graphing Sine and Cosine

Finally, you must be able to recognize and sketch the basic periodic graphs for $\sin\theta$ and $\cos\theta$.

• Sine Graph: Typically starts at the origin $(0,0)$ and follows a wave pattern that oscillates between $1$ and $-1$.
• Cosine Graph: Typically starts at its maximum value $(0,1)$ and follows a similar wave pattern.

Understanding the period and amplitude of these waves is essential for solving problems involving periodic motion or wave functions.

Final Study Advice: Do not just look at these formulas—write them out from memory multiple times. The midterm will require you to apply these identities to simplify expressions, so practice the algebraic steps of substituting one identity into another.