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In Exercises 8–12, draw each angle in standard position. 8𝜋 3
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 9
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.
cos 30°
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Identify the angle for which you need to find the cosine value, which is 30° in this case.
Recall that cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
From the triangle, note that the side adjacent to the 30° angle is \(\sqrt{3}\) and the hypotenuse is 2.
Set up the cosine ratio for 30°: \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
Since the denominator is already rational, no further rationalization is needed.
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Trigonometric Ratios in Right Triangles
Trigonometric ratios such as sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. For example, cosine of an angle is the ratio of the adjacent side to the hypotenuse. These ratios help evaluate expressions involving angles and side lengths.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Special Right Triangles (30°-60°-90° Triangle)
A 30°-60°-90° triangle has side lengths in a fixed ratio: the side opposite 30° is half the hypotenuse, and the side opposite 60° is √3 times the shorter leg. This property allows quick determination of side lengths and trigonometric values without complex calculations.
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Percorso guidato
04:3945-45-90 Triangles
Rationalizing the Denominator
Rationalizing the denominator involves eliminating 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
2:58Rationalizing Denominators
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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.
cos 2𝜋/3
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Domanda del libro di testo
In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋
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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 30°
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Domanda del libro di testo
In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 6 yards Arc Length, s: 8 yards
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Domanda del libro di testo
In Exercises 8–13, find the exact value of each expression. Do not use a calculator. tan 300°
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