Find values of the sine and cosine functions for each angle measure.

2x, given tan x = 5/3 and sin x < 0

Guida verificata passo dopo passo

Identify the given information: \( \tan x = \frac{5}{3} \) and \( \sin x < 0 \). This tells us the tangent ratio and the sign of the sine function, which helps determine the quadrant of angle \( x \).

Recall that \( \tan x = \frac{\sin x}{\cos x} \). Since \( \tan x = \frac{5}{3} \), we can represent \( \sin x = 5k \) and \( \cos x = 3k \) for some positive constant \( k \).

Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to find \( k \). Substitute \( \sin x = 5k \) and \( \cos x = 3k \) into the identity: \( (5k)^2 + (3k)^2 = 1 \).

Solve for \( k \) from the equation \( 25k^2 + 9k^2 = 1 \), which simplifies to \( 34k^2 = 1 \), then find \( k = \pm \frac{1}{\sqrt{34}} \).

Determine the correct sign of \( k \) using the condition \( \sin x < 0 \). Since \( \sin x = 5k \), \( k \) must be negative. Then calculate \( \sin 2x \) and \( \cos 2x \) using the double-angle formulas: \( \sin 2x = 2 \sin x \cos x \) and \( \cos 2x = \cos^2 x - \sin^2 x \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Trigonometric Ratios and Their Relationships

Trigonometric ratios (sine, cosine, tangent) relate the angles of a right triangle to the ratios of its sides. Knowing one ratio, such as tangent, allows calculation of sine and cosine using identities or the Pythagorean theorem, essential for finding values of sine and cosine from tangent.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2x in terms of functions of x. For sine and cosine, these are sin(2x) = 2 sin x cos x and cos(2x) = cos² x - sin² x, enabling calculation of sine and cosine for 2x once sin x and cos x are known.

Video consigliato:

Percorso guidato

05:06

Double Angle Identities

Sign Determination Based on Quadrants

The signs of sine, cosine, and tangent depend on the angle's quadrant. Given sin x < 0 and tan x = 5/3, the angle x lies in the third quadrant where sine and cosine are negative, but tangent is positive. This information is crucial to assign correct signs to sine and cosine values.

Video consigliato:

Percorso guidato