Textbook Question
Perform each calculation. See Example 3. 90° ― 51° 28'
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Ch. 1 - Trigonometric Functions
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 2, Problem 43
Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.
sec θ = 5/4 , and θ is in quadrant IV
Verified step by step guidance1
Recall that the secant function is the reciprocal of the cosine function, so from \(\sec \theta = \frac{5}{4}\), we can find \(\cos \theta\) by taking the reciprocal: \(\cos \theta = \frac{4}{5}\).
Since \(\theta\) is in quadrant IV, cosine is positive and sine is negative in this quadrant. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\). Substitute \(\cos \theta = \frac{4}{5}\) and solve for \(\sin \theta\).
Determine the sign of \(\sin \theta\) based on the quadrant. Since \(\theta\) is in quadrant IV, \(\sin \theta\) should be negative. So, take the negative root from the previous step.
Find \(\tan \theta\) using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) found earlier.
Calculate the remaining reciprocal functions: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{1}{\tan \theta}\). Rationalize denominators if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Trigonometric Functions
The six trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. Secant (sec θ) is the reciprocal of cosine (cos θ), meaning sec θ = 1/cos θ. Knowing sec θ allows you to find cos θ, which is essential for determining the other functions.
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Introduction to Trigonometric Functions
Sign of Trigonometric Functions in Quadrants
The sign of trigonometric functions depends on the quadrant of the angle. In quadrant IV, cosine and secant are positive, while sine and cosecant are negative. This knowledge helps assign correct signs to the calculated values of all six functions.
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Pythagorean Identity and Rationalizing Denominators
The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of sine once cosine is known. Rationalizing denominators involves rewriting expressions to eliminate radicals or fractions in the denominator, ensuring answers are in standard simplified form.
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Related Practice
Textbook Question
Perform each calculation.
110° 25' + 32° 55'
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Textbook Question
Perform each calculation. See Example 3. 47° 29' ― 71° 18'
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Textbook Question
Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3.
tan θ < 0 , cos θ < 0
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Textbook Question
Concept Check Classify each triangle as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.
Textbook Question
Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) III , r/y
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