Textbook Question
In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places.
tan 32.7°
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 41
In Exercises 41–43, find the exact value of each of the remaining trigonometric functions of θ.
cos θ = 2/5, sin θ < 0
Verified step by step guidance1
Identify the given information: \(\cos \theta = \frac{2}{5}\) and \(\sin \theta < 0\). This tells us the cosine value and that the sine is negative, which helps determine the quadrant where \(\theta\) lies.
Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\). Use this to find \(\sin \theta\) by substituting \(\cos \theta = \frac{2}{5}\) into the equation.
Calculate \(\sin^{2} \theta = 1 - \cos^{2} \theta = 1 - \left(\frac{2}{5}\right)^{2} = 1 - \frac{4}{25} = \frac{21}{25}\). Then, \(\sin \theta = \pm \sqrt{\frac{21}{25}} = \pm \frac{\sqrt{21}}{5}\).
Since \(\sin \theta < 0\), choose the negative value: \(\sin \theta = -\frac{\sqrt{21}}{5}\).
Find the remaining trigonometric functions using the definitions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) to express each function exactly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Pythagorean Identity
The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. Given cos θ, this identity allows you to find sin θ by rearranging the equation. Since cos θ = 2/5, you can calculate sin θ as ±√(1 - (2/5)²), choosing the sign based on the given condition.
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Pythagorean Identities
Sign of Trigonometric Functions in Quadrants
The sign of sine and cosine depends on the quadrant where the angle θ lies. Since sin θ < 0 and cos θ = 2/5 (positive), θ must be in the fourth quadrant, where cosine is positive and sine is negative. This helps determine the correct sign for sin θ and other functions.
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Definition of Trigonometric Functions
Trigonometric functions such as tangent, cotangent, secant, and cosecant are defined in terms of sine and cosine. For example, tan θ = sin θ / cos θ, sec θ = 1 / cos θ, etc. Once sin θ and cos θ are known, these remaining functions can be calculated exactly.
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Related Practice
Textbook Question
Find the reference angle for each angle.
5π/4
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Textbook Question
In Exercises 35–60, find the reference angle for each angle.
7𝜋/4
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Textbook Question
In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.
cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)
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Textbook Question
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.
7𝜋/6
Textbook Question
In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places.
csc 17°
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