Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 15

Chapter 1, Problem 15

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

sin 𝜋/4 - cos 𝜋/4

Verified step by step guidance

Identify the given triangle as a 45°-45°-90° right triangle, where the legs are equal and the hypotenuse is \(\sqrt{2}\) times the length of each leg.

Recall the definitions of sine and cosine for angle \(\pi/4\) (which is 45°): \(\sin(\pi/4) = \frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos(\pi/4) = \frac{\text{adjacent}}{\text{hypotenuse}}\).

Using the triangle, find \(\sin(\pi/4)\) as \(\frac{1}{\sqrt{2}}\) and \(\cos(\pi/4)\) as \(\frac{1}{\sqrt{2}}\) because both legs are 1 and the hypotenuse is \(\sqrt{2}\).

Set up the expression \(\sin(\pi/4) - \cos(\pi/4)\) and substitute the values found: \(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\).

Simplify the expression and if necessary, rationalize the denominator by multiplying numerator and denominator by \(\sqrt{2}\) to eliminate the square root in the denominator.

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Key Concepts

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

45°-45°-90° Triangle Properties

A 45°-45°-90° triangle is an isosceles right triangle where the legs are congruent, and the hypotenuse is √2 times the length of each leg. This relationship helps in determining side lengths and trigonometric ratios for angles of 45°.

Trigonometric Ratios for 45°

For a 45° angle in a right triangle, both sine and cosine values are equal because the legs opposite and adjacent to the angle are the same length. Specifically, sin(45°) = cos(45°) = √2/2, which simplifies calculations involving these angles.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This process simplifies expressions and is often required for final answers in trigonometry problems.

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Rationalizing Denominators

Related Practice

Textbook Question

In Exercises 13–17, find a positive angle less than 360° or 2𝜋 that is coterminal with the given angle. -445°

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Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.6 3 2 3 6 6 3 2 3 6Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.cos 3𝜋/2

Textbook Question

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

tan 𝜋/4 + csc 𝜋/6

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Textbook Question

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.sec 5𝜋/3

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Textbook Question

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.tan 3𝜋/2

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Textbook Question

The unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

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sin 3𝜋/2

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.sin 𝜋/4 - cos 𝜋/4

Verified video answer for a similar problem:

Key Concepts

45°-45°-90° Triangle Properties

Trigonometric Ratios for 45°

Rationalizing the Denominator

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

sin 𝜋/4 - cos 𝜋/4