Skip to main content
Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblem 3.3.55a
Chapter 3, Problem 3.3.55a

In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
tan α = 4/3, 180° < α < 270°

Verified step by step guidance
1
Identify the quadrant where the angle \( \alpha \) lies. Since \( 180^\circ < \alpha < 270^\circ \), \( \alpha \) is in the third quadrant, where both sine and cosine values are negative, but tangent is positive.
Recall the given value: \( \tan \alpha = \frac{4}{3} \). Use the identity \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) to relate sine and cosine.
Set \( \sin \alpha = 4k \) and \( \cos \alpha = 3k \) for some constant \( k \), because the ratio of sine to cosine must be \( \frac{4}{3} \). Since \( \alpha \) is in the third quadrant, both sine and cosine are negative, so \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \).
Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \): \[ (-4k)^2 + (-3k)^2 = 1 \\ 16k^2 + 9k^2 = 1 \\ 25k^2 = 1 \\ k^2 = \frac{1}{25} \\ k = \frac{1}{5} \]
Substitute \( k = \frac{1}{5} \) back into \( \sin \alpha = -4k \) to find \( \sin \alpha \): \[ \sin \alpha = -4 \times \frac{1}{5} = -\frac{4}{5} \]

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Ratios and Their Signs in Quadrants

Trigonometric functions like sine and tangent have specific signs depending on the quadrant of the angle. Since 180° < α < 270°, α lies in the third quadrant where sine is negative and tangent is positive. Understanding these sign rules is essential to determine the correct value of sin(α/2).
Recommended video:
6:36
Quadratic Formula

Half-Angle Formulas

Half-angle formulas allow calculation of trigonometric values for half of a given angle. For sine, the formula is sin(α/2) = ±√((1 - cos α)/2). Choosing the correct sign depends on the quadrant of α/2, which must be determined from the original angle α.
Recommended video:
6:36
Quadratic Formula

Relationship Between Tangent and Cosine

Given tan α, you can find cos α using the identity tan α = sin α / cos α and the Pythagorean identity sin² α + cos² α = 1. This step is crucial because the half-angle formula for sine requires the value of cos α.
Recommended video:
5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°
Related Practice
Textbook Question

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. cos² 15° - sin² 15°

Textbook Question

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x = 2 cos x tan x

2
views
Textbook Question

In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x

6
views
Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅

Textbook Question

In Exercises 55–58, use the given information to find the exact value of each of the following:

a. sin(α/2)

sec α = ﹣3, 𝝅/2 < α < 𝝅

4
views
Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.