Question

1. the point $(-4,3)$ lies on the terminal arm of a rotation angle as shown. determine the primary and reciprocal trigonometric ratios for the rotation angle. express each answer as an exact value.

Explanation:

Step1: Calculate the radius \( r \)

Use the formula \( r=\sqrt{x^{2}+y^{2}} \). Given \( x = - 4\) and \( y = 3\), then \( r=\sqrt{(-4)^{2}+3^{2}}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step2: Calculate the trigonometric ratios

• Sine: \( \sin\theta=\frac{y}{r}=\frac{3}{5}\), and its reciprocal \( \csc\theta=\frac{r}{y}=\frac{5}{3}\).
• Cosine: \( \cos\theta=\frac{x}{r}=\frac{-4}{5}\), and its reciprocal \( \sec\theta=\frac{r}{x}=-\frac{5}{4}\).
• Tangent: \( \tan\theta=\frac{y}{x}=\frac{3}{-4}=-\frac{3}{4}\), and its reciprocal \( \cot\theta=\frac{x}{y}=-\frac{4}{3}\).

Answer:

\(\sin\theta=\frac{3}{5},\csc\theta=\frac{5}{3},\cos\theta=-\frac{4}{5},\sec\theta=-\frac{5}{4},\tan\theta=-\frac{3}{4},\cot\theta=-\frac{4}{3}\)