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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 6
Chapter 3, Problem 6

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.
tan 2α

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1
Identify the given right triangle with sides: opposite = 11, adjacent = 60, hypotenuse = 61.
Use the tangent function for angle \( \alpha \): \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{11}{60} \).
Apply the double angle formula for tangent: \( \tan 2\alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} \).
Substitute \( \tan \alpha = \frac{11}{60} \) into the double angle formula.
Simplify the expression to find \( \tan 2\alpha \).

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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, tangent is the ratio of the opposite side to the adjacent side. Understanding these functions is essential for solving problems involving angles and side lengths in triangles.
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Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For tangent, the formula is tan(2α) = 2tan(α) / (1 - tan²(α)). These formulas are useful for simplifying expressions and solving trigonometric equations, especially when dealing with angles that are multiples of a given angle.
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Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is fundamental in trigonometry as it allows for the calculation of side lengths when angles are known, and vice versa. In the given triangle, it can be used to verify the relationships between the sides and to find missing values.
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Related Practice
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 7–14, use the given information to find the exact value of each of the following:b. cos 2θ15sin θ = -------- , θ lies in quadrant II.17
Textbook Question
In Exercises 1–60, verify each identity.tan x csc x cos x = 1
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Textbook Question
In Exercises 7–14, use the given information to find the exact value of each of the following:a. sin 2θ15sin θ = -------- , θ lies in quadrant II.17
Textbook Question

Use substitution to determine whether the given x-value is a solution of the equation.

tan2x=−33,x=5π12\(\tan\) 2x = -\(\frac{\sqrt{3}\)}{3}, \(\quad\) x = \(\frac{5\pi}{12}\)

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