AP Pre-Calculus: Trigonometric Identities - Your Ultimate Study Guide 🚀

Hey there! Let's get you prepped and confident for your AP Pre-Calculus exam. This guide is designed to be your go-to resource, especially the night before the test. We'll break down trigonometric identities into manageable, easy-to-remember chunks. Let's do this! 💪

Fundamental Trigonometric Identities

The Pythagorean Identity

It states that for any angle *x*:

$$sin^2(x) + cos^2(x) = 1$$

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This identity comes directly from the Pythagorean theorem applied to the unit circle. Think of it as the backbone of many other trig identities. 💡

Key Manipulations:

• $$cos^2(x) = 1 - sin^2(x)$$

• $$sin^2(x) = 1 - cos^2(x)$$

Example: Simplify $$2sin^2(x) + 2cos^2(x) - 1$$

1. Factor out the 2: $$2(sin^2(x) + cos^2(x)) - 1$$

2. Apply the Pythagorean identity: $$2(1) - 1$$

3. Simplify: $$2 - 1 = 1$$

More Pythagorean Identities

These are derived from the basic Pythagorean identity:

1. Tangent and Secant:

$$1 + tan^2(x) = sec^2(x)$$

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Derivation:

• Start with: $$tan(x) = \frac{sin(x)}{cos(x)}$$ and $$sec(x) = \frac{1}{cos(x)}$$
• Square the tangent: $$tan^2(x) = \frac{sin^2(x)}{cos^2(x)}$$
• Use $$sin^2(x) = 1 - cos^2(x)$$: $$tan^2(x) = \frac{1 - cos^2(x)}{cos^2(x)}$$
• Simplify: $$tan^2(x) = \frac{1}{cos^2(x)} - 1 = sec^2(x) - 1$$

2. Cotangent and Cosecant:

$$1 + cot^2(x) = csc^2(x)$$

Image courtesy of CollegeBoard.

Sum and Difference Identities

Sum Identities

These identities help you find trig values of sums of angles:

1. Sine Sum Identity:

$$sin(a + b) = sin(a)cos(b) + cos(a)sin(b)$$

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2. Cosine Sum Identity:

$$cos(a + b) = cos(a)cos(b) - sin(a)sin(b)$$

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Example: Find $$sin(15 + 30)$$

• $$sin(15 + 30) = sin(15)cos(30) + cos(15)sin(30)$$
• $$sin(15 + 30) = 0.2588 * 0.8660 + 0.9659 * 0.5$$
• $$sin(15 + 30) \approx 0.766$$

Difference Identities

Similar to sum identities, but for subtraction:

1. Sine Difference Identity:

$$sin(a - b) = sin(a)cos(b) - cos(a)sin(b)$$

2. Cosine Difference Identity:

$$cos(a - b) = cos(a)cos(b) + sin(a)sin(b)$$

Double-Angle Identities

These are special cases of the sum identities when the two angles are equal:

1. Sine Double-Angle Identity:

$$sin(2a) = 2sin(a)cos(a)$$

2. Cosine Double-Angle Identities:

$$cos(2a) = cos^2(a) - sin^2(a)$$

$$cos(2a) = 1 - 2sin^2(a)$$

$$cos(2a) = 2cos^2(a) - 1$$

Final Exam Focus 🎯

• High-Priority Topics: Pythagorean identities, sum and difference identities, and double-angle identities. These are foundational and appear frequently.
• Common Question Types: Simplifying expressions, solving equations, and proving identities. Expect to see a mix of these.
• Time Management: Practice recognizing patterns and applying identities quickly. Don't get bogged down on one problem.
• Common Pitfalls: Forgetting domain restrictions, mixing up signs in sum/difference identities, and not choosing the right form of the double-angle identity.
• Strategies: Start with the basic identities, manipulate them to match the problem, and always double-check your work.

Practice Questions

Practice Question

Multiple Choice Questions

1. What is the value of $$sin(2x - 30)$$ if $$sin(x) = 0.4$$ and $$cos(x) = 0.9$$?

   a) 0.76

   b) 0.24

   c) -0.24

   d) -0.76

2. Simplify $$tan(2x)$$ if $$sin(x) = 0.6$$ and $$cos(x) = 0.8$$

   a) 0.75

   b) 1.5

   c) 2.0

   d) 3.0

3. Simplify $$cos(a + b) - cos(a - b)$$

   a) $$2sin(a)sin(b)$$

   b) $$2cos(a)cos(b)$$

   c) $$sin(a+b) - sin(a-b)$$

   d) $$cos(a+b) + cos(a-b)$$

Answers:

1. c) -0.24
2. b) 1.5
3. a) $$2sin(a)sin(b)$$

Free Response Question

Given that $$sin(x) = \frac{3}{5}$$ and $$x$$ is in the first quadrant:

a) Find the value of $$cos(x)$$.

b) Find the value of $$sin(2x)$$.

c) Find the value of $$cos(2x)$$.

d) Find the value of $$tan(2x)$$.

Scoring Breakdown:

a) (2 points) Using the Pythagorean identity: $$cos^2(x) = 1 - sin^2(x)$$, $$cos^2(x) = 1 - (\frac{3}{5})^2 = \frac{16}{25}$$, $$cos(x) = \frac{4}{5}$$

b) (2 points) Using the double-angle identity: $$sin(2x) = 2sin(x)cos(x) = 2(\frac{3}{5})(\frac{4}{5}) = \frac{24}{25}$$

c) (3 points) Using the double-angle identity: $$cos(2x) = cos^2(x) - sin^2(x) = (\frac{4}{5})^2 - (\frac{3}{5})^2 = \frac{7}{25}$$ (or any other valid form)

d) (3 points) $$tan(2x) = \frac{sin(2x)}{cos(2x)} = \frac{24/25}{7/25} = \frac{24}{7}$$

You've got this! Keep practicing, and you'll ace that exam. Good luck! 🎉