Domanda del libro di testo
A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 1, Problema 2
Find the length of the arc on a circle of radius 20 feet intercepted by a 75° central angle. Express arc length in terms of 𝜋. Then round your answer to two decimal places.
Guida verificata passo dopo passo1
Identify the given values: the radius \(r = 20\) feet and the central angle \(\theta = 75^\circ\).
Recall the formula for the length of an arc: \(L = r \times \theta\), where \(\theta\) must be in radians.
Convert the central angle from degrees to radians using the conversion \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\), so calculate \(75^\circ \times \frac{\pi}{180}\).
Substitute the radius and the radian measure of the angle into the arc length formula: \(L = 20 \times \left(75 \times \frac{\pi}{180}\right)\).
Simplify the expression to write the arc length in terms of \(\pi\), then use a calculator to approximate the decimal value and round it to two decimal places.
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Arc Length Formula
The arc length of a circle is the distance along the curved line forming part of the circle's circumference. It is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula connects linear and angular measurements on a circle.
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Percorso guidato
4:18
Finding Missing Side Lengths
Conversion Between Degrees and Radians
Angles can be measured in degrees or radians. Since the arc length formula requires the angle in radians, convert degrees to radians by multiplying by π/180. For example, 75° equals (75 × π/180) radians, which is essential for accurate calculation.
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5:04Converting between Degrees & Radians
Rounding and Expressing Answers in Terms of π
Expressing the arc length in terms of π keeps the answer exact and symbolic. After finding the exact value, you can approximate the decimal value by substituting π ≈ 3.1416 and rounding to the desired decimal places, such as two decimals, for practical use.
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5:02Multiplying Complex Numbers
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In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.
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0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.
a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
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sin 3𝜋/4
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