Use identities to correctly complete each sentence.

If cos θ = 0.8 and sin θ = 0.6, then tan(-θ) = ________________.

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Recall the identity for tangent of a negative angle: \(\tan(-\theta) = -\tan(\theta)\).

Calculate \(\tan(\theta)\) using the given values of sine and cosine: \(\tan(\theta) = \frac{\sin \theta}{\cos \theta}\).

Substitute the given values into the tangent formula: \(\tan(\theta) = \frac{0.6}{0.8}\).

Apply the negative angle identity to find \(\tan(-\theta)\): \(\tan(-\theta) = -\frac{0.6}{0.8}\).

Simplify the fraction if needed to express the final answer in simplest form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. The primary ratios are sine (sin), cosine (cos), and tangent (tan), where tan θ = sin θ / cos θ. Understanding these ratios is essential for calculating one function from others.

Even-Odd Identities

Even-odd identities describe how trigonometric functions behave with negative angles. Cosine is an even function, so cos(-θ) = cos θ, while sine and tangent are odd functions, meaning sin(-θ) = -sin θ and tan(-θ) = -tan θ. These identities help evaluate functions at negative angles.

Using Given Values to Find Tangent

Given values of sine and cosine allow direct calculation of tangent using tan θ = sin θ / cos θ. With sin θ = 0.6 and cos θ = 0.8, tan θ = 0.6 / 0.8 = 0.75. Applying the even-odd identity for tangent then gives tan(-θ) = -0.75.

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