Question

find bc and then find the sine, cosine, and tangent of ∠a. give your answer as a fraction in simplest form. sin a= select cos a= select tan a= select (see yellow hint below.)

Explanation:

Step1: Find BC using Pythagorean theorem

In a right - triangle \(ABC\) with \(AC = 20\) (hypotenuse) and \(AB=16\), by the Pythagorean theorem \(BC=\sqrt{AC^{2}-AB^{2}}\).

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Step2: Recall trigonometric ratios

For \(\angle A\) in right - triangle \(ABC\):

• Sine of an angle \(\sin A=\frac{\text{opposite}}{\text{hypotenuse}}\). The side opposite to \(\angle A\) is \(BC = 12\) and the hypotenuse \(AC = 20\). So \(\sin A=\frac{BC}{AC}=\frac{12}{20}=\frac{3}{5}\).
• Cosine of an angle \(\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}\). The side adjacent to \(\angle A\) is \(AB = 16\) and the hypotenuse \(AC = 20\). So \(\cos A=\frac{AB}{AC}=\frac{16}{20}=\frac{4}{5}\).
• Tangent of an angle \(\tan A=\frac{\text{opposite}}{\text{adjacent}}\). The side opposite to \(\angle A\) is \(BC = 12\) and the side adjacent to \(\angle A\) is \(AB = 16\). So \(\tan A=\frac{BC}{AB}=\frac{12}{16}=\frac{3}{4}\).

Answer:

\(\sin A=\frac{3}{5}\), \(\cos A=\frac{4}{5}\), \(\tan A=\frac{3}{4}\)