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In Exercises 35–60, find the reference angle for each angle. -13𝜋/3
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.3.75
Blitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.3.75Capitolo 1, Problema 1.3.75
In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. tan(-𝜋/4)
Guida verificata passo dopo passo1
Recall that the tangent function is periodic and odd, meaning that \(\tan(-\theta) = -\tan(\theta)\). This property will help simplify \(\tan(-\pi/4)\).
Identify the reference angle for \(-\pi/4\). Since the angle is negative, its reference angle is the positive acute angle \(\pi/4\).
Find the value of \(\tan(\pi/4)\). From the unit circle or special triangles, \(\tan(\pi/4) = 1\).
Apply the odd function property: \(\tan(-\pi/4) = -\tan(\pi/4) = -1\).
Thus, the exact value of \(\tan(-\pi/4)\) is \(-1\).
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Reference Angles
A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to a corresponding angle in the first quadrant, where trigonometric values are well-known.
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Percorso guidato
5:31
Reference Angles on the Unit Circle
Tangent Function and Its Properties
The tangent of an angle in the unit circle is the ratio of the sine to the cosine of that angle. It is periodic with period π and odd, meaning tan(-θ) = -tan(θ), which is useful for evaluating tangent of negative angles without a calculator.
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Percorso guidato
5:43Introduction to Tangent Graph
Exact Values of Special Angles
Certain angles like π/4, π/3, and π/6 have known exact trigonometric values. For π/4, tan(π/4) = 1, so using these exact values avoids approximation and calculator use, enabling precise answers for trigonometric expressions.
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04:3945-45-90 Triangles
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Domanda del libro di testo
In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 17𝜋 /5
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Domanda del libro di testo
In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.
-210°
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Domanda del libro di testo
In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. tan θ = 0.4169
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Domanda del libro di testo
In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -𝜋/40
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Domanda del libro di testo
In Exercises 75–78, find the area of the sector of a circle of radius r formed by a central angle θ. Express area in terms of π. Then round your answer to two decimal places. Radius, r: 10 meters Central Angle, θ: θ = 18°
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