In Exercises 57–64, find the exact value of the following under the given conditions:
c. tan (α + β)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

검증된 단계별 안내

Identify the given values: \(\sin \alpha = \frac{3}{5}\) with \(\alpha\) in quadrant I, and \(\sin \beta = \frac{5}{13}\) with \(\beta\) in quadrant II.

Use the Pythagorean identity to find \(\cos \alpha\) and \(\cos \beta\). Since \(\sin^2 \theta + \cos^2 \theta = 1\), calculate \(\cos \alpha = \sqrt{1 - \sin^2 \alpha}\) and \(\cos \beta = -\sqrt{1 - \sin^2 \beta}\) (negative because \(\beta\) is in quadrant II where cosine is negative).

Calculate \(\tan \alpha\) and \(\tan \beta\) using the definitions \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) with the values found in the previous step.

Apply the tangent addition formula: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\).

Substitute the values of \(\tan \alpha\) and \(\tan \beta\) into the formula and simplify the expression to find the exact value of \(\tan(\alpha + \beta)\).

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Trigonometric Ratios and Quadrants

Trigonometric ratios like sine, cosine, and tangent relate the angles of a triangle to side lengths. Knowing the quadrant of an angle helps determine the sign (positive or negative) of these ratios, as sine is positive in quadrants I and II, cosine is positive in I and IV, and tangent's sign depends on sine and cosine.

Sum of Angles Formula for Tangent

The tangent of a sum of two angles, tan(α + β), can be found using the formula tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula allows calculation of the tangent of combined angles from the tangents of individual angles.

Finding Missing Trigonometric Ratios Using Pythagorean Identity

Given sin α or sin β, other ratios like cosine and tangent can be found using the Pythagorean identity sin²θ + cos²θ = 1. By solving for cosine and considering the quadrant, one can determine the correct sign and then compute tangent as sin θ / cos θ.

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.