Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
3
views
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Not the one you use?Change textbookSelect a different textbook
Table of contents
Chapter 3, Problem 3.RE.39
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. cos² 15° - sin² 15°
Verified step by step guidance1
Recognize that the expression \( \cos^2 15^\circ - \sin^2 15^\circ \) matches the form of the cosine double-angle identity, which states: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
Identify \( \theta = 15^\circ \) in the given expression, so the expression simplifies to \( \cos 2 \times 15^\circ = \cos 30^\circ \).
Recall the exact value of \( \cos 30^\circ \), which can be found using special right triangles or known trigonometric values.
Use the known exact value of \( \cos 30^\circ \) to express the final answer in simplest radical form.
Write the final exact value of the original expression \( \cos^2 15^\circ - \sin^2 15^\circ \) using the result from the double-angle formula.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Formulas
Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. For cosine, the formula cos(2θ) = cos²θ - sin²θ directly relates to the given expression, allowing simplification by recognizing the pattern.
Recommended video:
05:06
Double Angle Identities
Exact Values of Special Angles
Certain angles like 15°, 30°, 45°, and 60° have known exact trigonometric values involving square roots. Knowing or deriving these values is essential to find precise results without approximations when evaluating expressions like cos(15°) or sin(15°).
Recommended video:
04:3945-45-90 Triangles
Half-Angle Formulas
Half-angle formulas allow calculation of trigonometric functions of half an angle using the functions of the full angle. They are useful for breaking down angles like 15° into 30°/2, facilitating the evaluation of sine and cosine values needed in the problem.
Recommended video:
6:36Quadratic Formula
Related Practice
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 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°
1
views
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.