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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 4
Capitolo 3, Problema 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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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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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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Double Angle Identities

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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Solving Right Triangles with the Pythagorean Theorem
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In Exercises 57–64, find the exact value of the following under the given conditions:

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c. tan(α/2)

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