Domanda del libro di testo
In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = 4/3, cos θ < 0
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 30b
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of
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.
cot 15𝜋/2
Guida verificata passo dopo passo1
Step 1: Recognize that the problem asks for \( \cot(15\pi/2) \). The cotangent function is defined as \( \cot t = \frac{\cos t}{\sin t} \).
Step 2: Use the periodicity of the cotangent function. Since cotangent has a period of \( \pi \), reduce \( 15\pi/2 \) by subtracting multiples of \( \pi \) to find an equivalent angle within the first cycle. Calculate \( 15\pi/2 - 7\pi = (15/2 - 7)\pi = (15/2 - 14/2)\pi = \pi/2 \). So, \( \cot(15\pi/2) = \cot(\pi/2) \).
Step 3: Identify the coordinates on the unit circle corresponding to \( t = \pi/2 \). From the figure, the coordinates are \( (0, 1) \), where \( x = \cos(\pi/2) = 0 \) and \( y = \sin(\pi/2) = 1 \).
Step 4: Calculate \( \cot(\pi/2) = \frac{\cos(\pi/2)}{\sin(\pi/2)} = \frac{0}{1} \).
Step 5: Interpret the result from step 4 to understand the behavior of cotangent at \( \pi/2 \) and relate it back to the original angle.
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Unit Circle and Coordinates
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t, measured in radians, and has coordinates (x, y) = (cos t, sin t). These coordinates are essential for evaluating trigonometric functions at specific angles.
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Percorso guidato
06:11
Introduction to the Unit Circle
Cotangent Function
The cotangent of an angle t, cot(t), is defined as the ratio of the cosine to the sine of that angle: cot(t) = cos(t)/sin(t). Using the coordinates from the unit circle, cot(t) can be found by dividing the x-coordinate by the y-coordinate of the corresponding point.
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Percorso guidato
5:37Introduction to Cotangent Graph
Periodicity of Trigonometric Functions
Trigonometric functions like sine, cosine, and cotangent are periodic, meaning their values repeat at regular intervals. For cotangent, the period is π, so cot(t + kπ) = cot(t) for any integer k. This property allows simplification of angles outside the standard interval by reducing them modulo the period.
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Percorso guidato
5:33Period of Sine and Cosine Functions
Pratica correlata
Domanda del libro di testo
In Exercises 31–38, find a cofunction with the same value as the given expression. sin 7°
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Domanda del libro di testo
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of
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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cot 𝜋/2
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Domanda del libro di testo
In Exercises 30–32, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.
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Domanda del libro di testo
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of
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.
<Image>
sin 47𝜋/4
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Domanda del libro di testo
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 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.
tan 𝜋
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