Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1/ tan² α + cot α tan α
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.1.32
Find the remaining five trigonometric functions of θ.
cos θ = 1/5, θ in quadrant I
Verified step by step guidance1
Identify the given information: \(\cos \theta = \frac{1}{5}\) and \(\theta\) is in Quadrant I, where all trigonometric functions are positive.
Use the Pythagorean identity to find \(\sin \theta\). Recall that \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = \frac{1}{5}\) into the identity: \(\sin^2 \theta + \left(\frac{1}{5}\right)^2 = 1\).
Solve for \(\sin \theta\): \(\sin^2 \theta = 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25}\). Then, \(\sin \theta = \sqrt{1 - \frac{1}{25}}\). Since \(\theta\) is in Quadrant I, take the positive root.
Find \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) found in previous steps.
Calculate the reciprocal functions: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Use the values obtained for \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\).
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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. Given cos θ, this identity allows us to find sin θ by rearranging the equation to sin θ = √(1 - cos²θ). This is essential for determining the sine value when cosine is known.
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Pythagorean Identities
Signs of Trigonometric Functions in Quadrants
The sign of trigonometric functions depends on the quadrant of the angle. Since θ is in quadrant I, all six trigonometric functions (sin, cos, tan, cot, sec, csc) are positive. This information helps assign the correct sign to the calculated values.
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6:36Quadratic Formula
Definitions of Trigonometric Functions
The six trigonometric functions are defined as ratios of sides in a right triangle or as reciprocal functions: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ / cos θ, cot θ = 1 / tan θ, sec θ = 1 / cos θ, and csc θ = 1 / sin θ. Knowing these definitions allows calculation of all functions once sin θ and cos θ are known.
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Related Practice
Textbook Question
Find sin θ.
cos θ = 5/6, θ in quadrant I
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Textbook Question
Identify the basic trigonometric function graphed, and determine whether it is even or odd.
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Textbook Question
Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
Textbook Question
Find sinθ.
csc θ = -8/5
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
cos θ (cos θ - sec θ)