Trigonometry Simplification and Values

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Questions and Answers

How do you simplify the expression `(1 - cos(θ))/sin(θ)` using trigonometric identities?

The expression can be simplified to tan(θ/2) using the half-angle identity for tangent.

Given `cos(θ)` is a specific value, how can you find `cos(2θ)` in the interval `π ≤ θ ≤ 3π/2`?

Use the double angle formula cos(2θ) = 2cos²(θ) - 1 to find the value based on cos(θ). Consider the signs based on the given interval.

What values of θ make the expression `(1 - cos(θ))/sin(θ)` undefined?

The expression is undefined when sin(θ) = 0, which occurs at θ = nπ, where n is any integer.

Explain the significance of the interval `π ≤ θ ≤ 3π/2` for trigonometric function values.

This interval lies in the third quadrant, where both sine and cosine are negative, impacting the signs of trigonometric values derived from θ. Signup and view all the answers

What does the secant function represent and how is `sec(2θ)` derived from `cos(2θ)`?

The secant function is the reciprocal of cosine, so <code>sec(2θ) = 1/cos(2θ)</code>, and can be derived once <code>cos(2θ)</code> is calculated. Signup and view all the answers

The expression `(1 - cos(θ))/sin(θ)` is undefined when `sin(θ) = 0`.

True Signup and view all the answers

Which of the following expressions represents a simplification of `(1 - cos(θ))/sin(θ)`?

tan(θ/2) Signup and view all the answers

What is the range of θ for which `cos(θ)` is negative?

π < θ < 2π Signup and view all the answers

The secant function, denoted as ______, is the reciprocal of cosine.

sec Signup and view all the answers

Match the following trigonometric identities with their corresponding values:

cos(2θ) = 2cos²(θ) - 1 sec(2θ) = 1/cos(2θ) tan(θ) = sin(θ)/cos(θ) sin²(θ) + cos²(θ) = 1 Signup and view all the answers

Study Notes

Trigonometric Simplification

The document covers simplification of trigonometric expressions, including expressions involving sine, cosine, and tangent functions.
It specifically focuses on using trigonometric identities for simplification.

Finding Trigonometric Values

The document covers finding the exact values for trigonometric functions of specific angles.
It also includes examples of finding values for cos(2θ) and sec(2θ) given a value for cos(θ).

Identifying Non-Permissible Values

The document provides instructions for finding values of θ where trigonometric expressions are undefined.
This involves identifying where the denominator of an expression is equal to zero, making the expression undefined.

Visual Aids and Examples

The document utilizes handwritten calculations and diagrams to aid in understanding trigonometric transformations and simplification techniques.
Handwritten diagrams and calculations often provide a visual representation to accompany the mathematical steps, enhancing the understanding of the material.

Trigonometric Function Simplification

Focuses on simplifying trigonometric expressions using identities.
Utilizes sine, cosine, and tangent functions.
Includes examples demonstrating how to simplify expressions involving these functions.

Finding Trigonometric Function Values

Demonstrates finding exact values for trigonometric functions for specific angles.
Includes examples with specific angles and given values of cosine.
Illustrates finding values for cos(2θ) and sec(2θ).