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15 Find sin 2x, cos 2x, and tan 2x if tanx = 8 sin 2x = cos2x - 0 tan 2x 0 - 8 X and x terminates in quadrant I. S

15 Find sin 2x, cos 2x, and tan 2x if tanx = 8 sin 2x = cos2x - 0 tan 2x 0 - 8 X and x terminates in quadrant I. S

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) if \(\tan x = \frac{15}{8}\) and \(x\) terminates in quadrant I. **Solution Steps:** 1. **Given Information:** - \(\tan x = \frac{15}{8}\) - \(x\) is in quadrant I. 2. **Trigonometric Identities:** - \(\sin 2x = 2 \sin x \cos x\) - \(\cos 2x = \cos^2 x - \sin^2 x\) - \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\) 3. **Determine \(\sin x\) and \(\cos x\):** - Use \(\tan x = \frac{\sin x}{\cos x}\). - From the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\). - Calculate \(\sin x\) and \(\cos x\) using the given \(\tan x\). 4. **Calculate \(\sin 2x\), \(\cos 2x\), \(\tan 2x\):** - Substitute \(\sin x\) and \(\cos x\) into the identities for double angles. 5. **Enter the Values:** - Input boxes for \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) are provided for users to check their answers. **Note:** - Ensure calculations are consistent with the fact that \(x\) is in quadrant I.
Transcribed Image Text:**Problem Statement:** Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) if \(\tan x = \frac{15}{8}\) and \(x\) terminates in quadrant I. **Solution Steps:** 1. **Given Information:** - \(\tan x = \frac{15}{8}\) - \(x\) is in quadrant I. 2. **Trigonometric Identities:** - \(\sin 2x = 2 \sin x \cos x\) - \(\cos 2x = \cos^2 x - \sin^2 x\) - \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\) 3. **Determine \(\sin x\) and \(\cos x\):** - Use \(\tan x = \frac{\sin x}{\cos x}\). - From the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\). - Calculate \(\sin x\) and \(\cos x\) using the given \(\tan x\). 4. **Calculate \(\sin 2x\), \(\cos 2x\), \(\tan 2x\):** - Substitute \(\sin x\) and \(\cos x\) into the identities for double angles. 5. **Enter the Values:** - Input boxes for \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) are provided for users to check their answers. **Note:** - Ensure calculations are consistent with the fact that \(x\) is in quadrant I.
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