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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 4
Chapter 3, Problem 4

In Exercises 1–6, use the figures to find the exact value of each trigonometric function.A right triangle labeled with sides 11, 60, and 61, and angle Δ, for trigonometric function exercises.
sin 2α

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1
Identify the given right triangle with sides 60, 11, and hypotenuse 61.
Recognize that the angle \( \alpha \) is opposite the side of length 11.
Use the double angle identity for sine: \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \).
Calculate \( \sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{11}{61} \).
Calculate \( \cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{60}{61} \).

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Key Concepts

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

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions include sine, cosine, and tangent, which are defined as ratios of the sides of a right triangle. For example, in a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is essential for solving problems involving angles and side lengths.
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Introduction to Trigonometric Functions

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For instance, the sine double angle formula states that sin(2α) = 2sin(α)cos(α). These formulas are useful for simplifying expressions and solving equations involving trigonometric functions, particularly when calculating values for angles that are multiples of a given angle.
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Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is crucial for finding missing side lengths in trigonometric problems and for verifying the relationships between the sides of a triangle.
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Related Practice
Textbook Question

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

c. tan (α + β)

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

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Textbook Question
Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.sin x cos 2x
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Textbook Question
In Exercises 1–60, verify each identity.tan x csc x cos x = 1
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Textbook Question

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

c. tan(α/2)

tan α = 4/3, 180° < α < 270°

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Textbook Question

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

c. tan(α/2)

sec α = ﹣3, 𝝅/2 < α < 𝝅

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Textbook Question
Use 105° = 135° - 30° to find the exact value of 105°.
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