Textbook Question
Identify the basic trigonometric function graphed, and determine whether it is even or odd.
<IMAGE>
Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
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 5.1.10
Use identities to correctly complete each sentence.
If sin θ = ⅔, then -sin(-θ) = ________________.
Verified step by step guidance1
Recall the identity for the sine of a negative angle: \(\sin(-\theta) = -\sin(\theta)\).
Apply this identity to rewrite \(-\sin(-\theta)\) as \(-(-\sin(\theta))\).
Simplify the expression: \(-(-\sin(\theta)) = \sin(\theta)\).
Since it is given that \(\sin(\theta) = \frac{2}{3}\), substitute this value into the expression.
Therefore, \(-\sin(-\theta) = \frac{2}{3}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sine Function and Its Values
The sine function relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. It is defined for all real numbers and can take values between -1 and 1. Knowing the sine value of an angle helps determine related trigonometric expressions.
Recommended video:
5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°
Odd Function Property of Sine
Sine is an odd function, meaning sin(-θ) = -sin(θ). This property allows us to simplify expressions involving negative angles by changing the sign of the sine value, which is crucial for evaluating expressions like -sin(-θ).
Recommended video:
06:19Even and Odd Identities
Trigonometric Identities and Simplification
Trigonometric identities are equations involving trig functions that hold true for all angle values. Using these identities, such as the odd function property, helps simplify and correctly complete expressions by substituting known values and reducing complexity.
Recommended video:
5:32Fundamental Trigonometric Identities
Related Practice
Textbook Question
Find the remaining five trigonometric functions of θ.
cos θ = -1/4, sin θ > 0
Textbook Question
Factor each trigonometric expression.
(tan x + cot x)² - (tan x - cot x)²
Textbook Question
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
csc 18°
Textbook Question
Determine whether the positive or negative square root should be selected.
sin (-10°) = ± √[(1 - cos (-20°))/2]
1
views
Textbook Question
Verify that each equation is an identity.
tan² α sin² α = tan² α + cos² α - 1