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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblem 11
Chapter 3, Problem 11

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.
Right triangle ABC with right angle at C, sides a=5, b=12, c unknown; find c and trig functions for angle B.
a = 5, b = 12

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1
Identify the sides of the right triangle ABC, where the right angle is at C. This means sides a and b are the legs, and side c is the hypotenuse opposite the right angle.
Use the Pythagorean theorem to find the length of the hypotenuse c: \(c^2 = a^2 + b^2\). Substitute the given values: \(c^2 = 5^2 + 12^2\).
Calculate \(c^2\) by squaring the given sides: \(c^2 = 25 + 144\). Then express \(c\) as \(c = \sqrt{169}\), but do not simplify further here.
To find the six trigonometric functions for angle B, first identify the sides relative to angle B: the side opposite B is a, the side adjacent to B is b, and the hypotenuse is c.
Write the six trigonometric functions in terms of a, b, and c: - \(\sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}\) - \(\cos B = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c}\) - \(\tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b}\) - \(\csc B = \frac{1}{\sin B} = \frac{c}{a}\) - \(\sec B = \frac{1}{\cos B} = \frac{c}{b}\) - \(\cot B = \frac{1}{\tan B} = \frac{b}{a}\) Substitute the known values for a, b, and c into these expressions and rationalize denominators if 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 the missing side length when two sides are known.
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Trigonometric Functions

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, these functions are defined using the lengths of the opposite, adjacent, and hypotenuse sides, providing exact values based on the triangle's dimensions.
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Rationalizing Denominators

Rationalizing denominators involves rewriting a fraction so that its denominator contains no irrational numbers, such as square roots. This is done by multiplying numerator and denominator by a suitable radical to simplify expressions and present answers in a standard, exact form.
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Related Practice
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Textbook Question

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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

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

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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.

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