AP Precalculus Section 3.12 Example: Verify a Pythagorean Trigonometric Identity (Example 1)

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

Verifying a Pythagorean trigonometric identity involves proving that one side of the identity is equal to the other side. Pythagorean identities are a special class of trigonometric identities that are derived from the Pythagorean theorem. The most common Pythagorean identity is:

\[ \sin^2(x) + \cos^2(x) = 1 \]

Here's a general approach to verify a Pythagorean trigonometric identity:

### Step 1: Identify the Given Identity

Identify the Pythagorean identity you want to verify. It typically looks like \( \text{Expression}_1 = \text{Expression}_2 \), and you need to prove that both sides are equal.

### Step 2: Work with One Side

Choose one side of the identity (either \( \text{Expression}_1 \) or \( \text{Expression}_2 \)) to work with. Let's assume you choose \( \text{Expression}_1 \).

### Step 3: Use Trigonometric Identities

Use known trigonometric identities to simplify the chosen side of the equation. Common trigonometric identities include:

- **Reciprocal Identities:**
\[ \csc(x) = \frac{1}{\sin(x)}, \quad \sec(x) = \frac{1}{\cos(x)}, \quad \cot(x) = \frac{1}{\tan(x)} \]

- **Quotient Identities:**
\[ \tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)} \]

- **Pythagorean Identities:**
\[ \sin^2(x) + \cos^2(x) = 1, \quad \tan^2(x) + 1 = \sec^2(x), \quad 1 + \cot^2(x) = \csc^2(x) \]

### Step 4: Simplify Both Sides

Simplify the chosen side using the trigonometric identities until you obtain a simplified expression.

### Step 5: Compare Both Sides

After simplifying the chosen side, compare it to the other side of the Pythagorean identity. If both sides are equal, you have successfully verified the identity.

### Step 6: Repeat if Necessary

If you chose \( \text{Expression}_1 \) initially, repeat the process by choosing \( \text{Expression}_2 \) and following the same steps to verify the identity from both perspectives.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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