Use identities to correctly complete each sentence.

If sin θ = ⅔, then -sin(-θ) = ________________.

Verified step by step guidance

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, \(-\sin(-\theta) = \frac{2}{3}\).

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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.

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).

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.

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