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: 1 meter Arc Length, s: 600 centimeters
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 13
Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
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tan 𝜋/3
Guida verificata passo dopo passo1
Recall that the angle \( \frac{\pi}{3} \) radians corresponds to 60 degrees, a common special angle in trigonometry.
Identify or recall the side lengths of a 30-60-90 right triangle, which are in the ratio 1 : \( \sqrt{3} \) : 2, where the side opposite 60 degrees (\( \frac{\pi}{3} \)) is \( \sqrt{3} \), the side opposite 30 degrees is 1, and the hypotenuse is 2.
Use the definition of tangent for an angle in a right triangle: \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \). For \( \theta = \frac{\pi}{3} \), the opposite side is \( \sqrt{3} \) and the adjacent side is 1.
Write the expression for \( \tan \frac{\pi}{3} \) as \( \frac{\sqrt{3}}{1} \).
Since the denominator is already rational (1), no rationalization is needed. The expression is simplified as is.
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Definition of the Tangent Function
The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side. For angle θ, tan(θ) = opposite/adjacent. This ratio helps evaluate trigonometric expressions using triangle side lengths.
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Percorso guidato
5:43
Introduction to Tangent Graph
Special Angles and Their Trigonometric Values
Certain angles like π/3 (60°) have well-known exact trigonometric values. For π/3, tan(π/3) equals √3. Recognizing these special angles allows quick evaluation without needing a calculator or complex calculations.
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Percorso guidato
3:28Common Trig Functions For 45-45-90 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.
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Percorso guidato
2:58Rationalizing Denominators
Pratica correlata
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.
sin 𝜋/4 - cos 𝜋/4
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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.
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In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.
sec 11𝜋/6
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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.
sec 5𝜋/3
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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. csc 4𝜋/3
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