Textbook Question
Simplify each expression.
√[(1 + cos 76°)/2]
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 34
Use identities to fill in each blank with the appropriate trigonometric function name.
____ 72° = cot 18°
Verified step by step guidance1
Recall the complementary angle identity for cotangent: \(\cot(\theta) = \tan(90^\circ - \theta)\).
Apply this identity to the given expression \(\cot 18^\circ\): rewrite it as \(\tan(90^\circ - 18^\circ)\).
Calculate the complementary angle: \(90^\circ - 18^\circ = 72^\circ\).
Therefore, \(\cot 18^\circ = \tan 72^\circ\).
Fill in the blank with the function name 'tan' to complete the identity: \(\tan 72^\circ = \cot 18^\circ\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complementary Angle Identities
Complementary angle identities relate trigonometric functions of angles that add up to 90°. For example, sin(90° - θ) = cos θ and tan(90° - θ) = cot θ. These identities help transform one trigonometric function into another by using the complementary angle.
Recommended video:
Guided course
3:35
Intro to Complementary & Supplementary Angles
Cotangent Function
The cotangent function, cot θ, is the reciprocal of the tangent function: cot θ = 1/tan θ. It can also be expressed as the ratio of the adjacent side to the opposite side in a right triangle. Understanding cotangent is essential for recognizing its relationship with other trig functions.
Recommended video:
5:37Introduction to Cotangent Graph
Using Identities to Solve for Unknown Functions
Trigonometric identities allow substitution and simplification of expressions involving trig functions. By applying identities like complementary angle relationships, one can find equivalent expressions for unknown functions, enabling the completion of equations or simplification of expressions.
Recommended video:
04:42Solve Trig Equations Using Identity Substitutions
Related Practice
Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
cos(45° - θ)
Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
cos(θ - 30°)
Textbook Question
Use identities to fill in each blank with the appropriate trigonometric function name.
sin 2π/3 = _____ (- π/6)
Textbook Question
If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .
Textbook Question
Simplify each expression.
√[(1 + cos 165°)/(1 - cos 165°)]