Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 13

Chapter 3, Problem 13

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. a = 6, c = 7

Verified step by step guidance

Identify the sides of the right triangle ABC, where the right angle is at C. This means side c is the hypotenuse, and sides a and b are the legs adjacent to angle C. Given a = 6 and c = 7, we need to find side b using the Pythagorean theorem.

Apply the Pythagorean theorem: \(a^2 + b^2 = c^2\). Substitute the known values: \(6^2 + b^2 = 7^2\).

Simplify the equation: \(36 + b^2 = 49\). Then solve for \(b^2\) by subtracting 36 from both sides: \(b^2 = 49 - 36\).

Calculate \(b\) by taking the square root of both sides: \(b = \sqrt{13}\). This gives the exact length of side b.

Next, find the six trigonometric functions for angle B. Recall that angle B is opposite side b, adjacent to side a, and the hypotenuse is c. Use the definitions: \(\sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c}\), \(\cos B = \frac{a}{c}\), \(\tan B = \frac{b}{a}\), and their reciprocals \(\csc B = \frac{c}{b}\), \(\sec B = \frac{c}{a}\), \(\cot B = \frac{a}{b}\). Substitute the known values and rationalize denominators where necessary.

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Key Concepts

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

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (side opposite the right angle) equals the sum of the squares of the other two sides. It is expressed as c² = a² + b². This theorem allows you to find an unknown side length when the other two are known.

Trigonometric Functions in Right Triangles

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—relate the angles of a right triangle to the ratios of its sides. For an angle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. The reciprocal functions are cosecant, secant, and cotangent.

Rationalizing Denominators

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable radical expression. It is a standard practice to present trigonometric values in simplified, exact form.

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Related Practice

Textbook Question

Determine whether each statement is true or false. If false, tell why. csc 22° ≤ csc 68°

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Textbook Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. b = 8, c = 11

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Textbook Question

Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. 480° F. 32°

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Textbook Question

Find exact values of the six trigonometric functions for each angle. Do not use a calculator. Rationalize denominators when applicable. 120°

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Textbook Question

Solve each right triangle. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.

a = 5, b = 12