Question

1. sketch the angle in standard position and find the exact value for each trigonometric function given the terminal side passes through (-3,-4).

Explanation:

Step1: Calculate the radius \(r\)

For a point \((x,y)=(-3,-4)\) on the terminal side of an angle \(\theta\) in standard position, use the formula \(r = \sqrt{x^{2}+y^{2}}\).

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Step2: Calculate the trigonometric functions

• Sine function: \(\sin\theta=\frac{y}{r}\), with \(y=-4\) and \(r = 5\), so \(\sin\theta=\frac{-4}{5}\)
• Cosecant function: \(\csc\theta=\frac{r}{y}\), with \(r = 5\) and \(y=-4\), so \(\csc\theta=\frac{5}{-4}=-\frac{5}{4}\)
• Cosine function: \(\cos\theta=\frac{x}{r}\), with \(x=-3\) and \(r = 5\), so \(\cos\theta=\frac{-3}{5}\)
• Secant function: \(\sec\theta=\frac{r}{x}\), with \(r = 5\) and \(x=-3\), so \(\sec\theta=\frac{5}{-3}=-\frac{5}{3}\)
• Tangent function: \(\tan\theta=\frac{y}{x}\), with \(y=-4\) and \(x=-3\), so \(\tan\theta=\frac{-4}{-3}=\frac{4}{3}\)
• Cotangent function: \(\cot\theta=\frac{x}{y}\), with \(x=-3\) and \(y=-4\), so \(\cot\theta=\frac{-3}{-4}=\frac{3}{4}\)

Answer:

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