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 3.2.29
Chapter 3, Problem 3.2.29

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.
29. sin(5𝝅/12) cos(𝝅/4) - cos(5𝝅/12) sin(𝝅/4)

Verified step by step guidance
1
Identify the given expression: \(\sin \frac{5\pi}{12} \cos \frac{\pi}{4} - \cos \frac{5\pi}{12} \sin \frac{\pi}{4}\).
Recognize that this expression matches the sine difference identity: \(\sin A \cos B - \cos A \sin B = \sin(A - B)\).
Assign \(A = \frac{5\pi}{12}\) and \(B = \frac{\pi}{4}\), then rewrite the expression as \(\sin\left( \frac{5\pi}{12} - \frac{\pi}{4} \right)\).
Calculate the difference inside the sine function: find a common denominator and subtract \(\frac{5\pi}{12} - \frac{\pi}{4}\).
Express the original expression as \(\sin\) of the resulting angle and then evaluate the exact value using known sine values or special angle properties.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m
Was this helpful?

Key Concepts

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

Sum and Difference Identities

Sum and difference identities express the sine, cosine, or tangent of sums or differences of angles in terms of the sines and cosines of the individual angles. For example, sin(a Β± b) = sin a cos b Β± cos a sin b. These identities allow rewriting complex trigonometric expressions as single trigonometric functions of combined angles.
Recommended video:
2:25
Verifying Identities with Sum and Difference Formulas

Exact Values of Trigonometric Functions

Certain angles, especially multiples of Ο€/6, Ο€/4, and Ο€/3, have known exact sine, cosine, and tangent values. Using these exact values avoids decimal approximations and provides precise results. Recognizing these angles helps in evaluating expressions after applying sum or difference identities.
Recommended video:
6:04
Introduction to Trigonometric Functions

Angle Conversion and Simplification

Understanding how to manipulate and simplify angles, such as converting fractions of Ο€ into recognizable standard angles, is essential. This includes reducing angles to their reference angles within the unit circle to apply known values and identities effectively, facilitating the evaluation of trigonometric expressions.
Recommended video:
04:46
Coterminal Angles
Related Practice
Textbook Question
In Exercises 1–60, verify each identity.tan (-x) cos x = -sin x
2
views
Textbook Question

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 cosΒ² x + 3 cos x + 1 = 0

Textbook Question

In Exercises 59–68, verify each identity.

cosΒ²(ΞΈ/2) = (sec ΞΈ + 1)/(2 sec ΞΈ)

1
views
Textbook Question

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sinΒ² x = sin x + 3

Textbook Question
In Exercises 1–6, use the figures to find the exact value of each trigonometric function.

tan 2ΞΈ
2
views
Textbook Question

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan(x/2) = √3

5
views