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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 15
Capitolo 1, Problema 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.
Right triangle ABC with angles 45°, sides 1, and hypotenuse √2.
sin 𝜋/4 - cos 𝜋/4

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1
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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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°.
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Percorso guidato
4:42
Review of Triangles

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.
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Percorso guidato
6:04
Introduction to Trigonometric Functions

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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Percorso guidato
2:58
Rationalizing Denominators
Pratica correlata
Domanda del libro di testo

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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Domanda del libro di testo
In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 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.

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

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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Domanda del libro di testo
In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 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.

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

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.


<IMAGE>


tan 𝜋/3

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Domanda del libro di testo

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.

<IMAGE>


sin 3𝜋/2

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