Chapter 7: Fundamental Trigonometric Identities and Their Applications

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Chapter 7: Trigonometric Identities and Equations

Fundamental Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable for which both sides are defined. These identities are essential tools for simplifying expressions and solving trigonometric equations.

Reciprocal Identities: These relate each trigonometric function to its reciprocal.

Reciprocal identities for trigonometric functions

Quotient Identities: These express tangent and cotangent in terms of sine and cosine.

Pythagorean Identities: These are derived from the Pythagorean Theorem and relate the squares of sine, cosine, and tangent.
Even-Odd Identities: These describe the symmetry properties of trigonometric functions with respect to the origin.

Pythagorean identities for trigonometric functions Even-odd identities for trigonometric functions

Understanding the Argument of Trigonometric Functions

In trigonometric identities, the argument (such as \( \theta \)) can represent an angle in degrees, a real number, or a variable. It is important to always specify the argument when writing trigonometric expressions to avoid ambiguity.

Evaluating Trigonometric Functions Given One Value and the Quadrant

When given the value of one trigonometric function and the quadrant in which the angle lies, you can use identities to find the values of the other trigonometric functions. The sign of each function depends on the quadrant.

Example 1: Given \( \tan \theta = -\frac{5}{3} \) and \( \theta \) is in quadrant II, find the values of the other trigonometric functions.

Given tan theta = -5/3, theta in quadrant II

(a) Finding \( \sec \theta \): Use the Pythagorean identity \( 1 + \tan^2 \theta = \sec^2 \theta \) to solve for \( \sec \theta \).

Pythagorean identity for secant Calculation for secant using Pythagorean identity Final value for secant

(b) Finding \( \sin \theta \): Use the reciprocal and quotient identities to express \( \sin \theta \) in terms of \( \tan \theta \) and \( \sec \theta \).

Quotient identity for tangent Reciprocal identity for secant Calculation for sine using secant and tangent Simplification for sine Final value for sine

(c) Finding \( \cot(-\theta) \): Use the reciprocal and negative-angle identities.

Reciprocal identity for cotangent Negative-angle identity for cotangent Calculation for cotangent of negative angle

Caution: When taking square roots, always choose the sign based on the quadrant of \( \theta \) and the function being evaluated.

Expressing One Trigonometric Function in Terms of Another

It is often useful to rewrite one trigonometric function in terms of another using identities. This is especially helpful when solving equations or simplifying expressions.

Example 2: Write \( \cos x \) in terms of \( \tan x \).

Pythagorean identity for tangent and secant Reciprocal identity for secant Solving for cosine in terms of tangent

The sign of the result depends on the quadrant of \( x \).

Rewriting Expressions in Terms of Sine and Cosine

Many trigonometric expressions can be simplified by rewriting all functions in terms of sine and cosine, then using algebraic techniques and identities to eliminate quotients.

Example 3: Rewrite an expression in terms of sine and cosine, and simplify so that no quotients appear.

Use quotient identities, multiply numerator and denominator by the least common denominator (LCD), and apply reciprocal and Pythagorean identities as needed.

Importance of Arguments in Trigonometric Identities

Always include the argument (such as \( \theta \)) when writing trigonometric identities. Omitting the argument can lead to confusion or incorrect statements.

Example showing the necessity of including the argument in trigonometric identities

Caution: When working with trigonometric expressions and identities, always write the argument of the function.

Practice Problems

Question 1: Find the five remaining trigonometric functions of \( \alpha \) given \( \tan \alpha = -\frac{1}{4} \), \( \alpha \) in quadrant IV.

Practice problem: Find remaining trig functions given tangent and quadrant

Question 2: Write each expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression and all functions are of \( \theta \) only.

Practice problem: Simplify trigonometric expression using identities

Additional info: The above notes are based on standard precalculus curriculum and the provided textbook images, focusing on Chapter 7: Trigonometric Identities and Equations.