Textbook Question
Determine whether each statement is true or false. If false, tell why. Use a calculator for Exercises 39 and 42. sin 42° + sin 42° = sin 84°
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Ch. 2 - Acute Angles and Right Triangles
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 3, Problem 2.R.25
Use a calculator to approximate the value of each expression. Give answers to six decimal places. sec 222° 30'
Verified step by step guidance1
Convert the angle from degrees and minutes to decimal degrees. Since 30' means 30 minutes, and 1 degree = 60 minutes, calculate the decimal degrees as \(222 + \frac{30}{60}\).
Recall that \(\sec \theta = \frac{1}{\cos \theta}\). So, to find \(\sec 222^\circ 30'\), you first need to find \(\cos 222.5^\circ\) (the decimal degree equivalent).
Use a calculator set to degree mode to find the value of \(\cos 222.5^\circ\).
Calculate the secant by taking the reciprocal of the cosine value: \(\sec 222.5^\circ = \frac{1}{\cos 222.5^\circ}\).
Round your final answer to six decimal places as requested.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Understanding the Secant Function
The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). To find secant values, you first calculate the cosine of the angle and then take its reciprocal. This relationship is fundamental when working with trigonometric expressions involving secant.
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Graphs of Secant and Cosecant Functions
Converting Degrees and Minutes to Decimal Degrees
Angles given in degrees and minutes must be converted to decimal degrees for calculator use. Since 1 minute equals 1/60 of a degree, convert 30' to 0.5°, so 222° 30' becomes 222.5°. This conversion ensures accurate input for trigonometric calculations on calculators.
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Using a Calculator for Trigonometric Approximations
Calculators typically require angles in decimal degrees or radians to compute trigonometric functions. After converting the angle, input it correctly, ensure the calculator is in degree mode, compute cos(θ), then find sec(θ) by taking the reciprocal. Finally, round the result to six decimal places as requested.
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Related Practice
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