Problem 2

Question

In Exercises $1-8,$ a point on the terminal side of angle $\theta$ is given. Find the exact value of each of the six trigonometric functions of $\theta$. $$(-12,5)$$

Step-by-Step Solution

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Answer

The exact values of the six trigonometric functions for the given point (-12,5) are: $sin\theta = \frac{5}{13}$, $cos\theta = \frac{-12}{13}$, $tan\theta = -\frac{5}{12}$, $csc\theta = \frac{13}{5}$, $sec\theta = -\frac{13}{12}$, $cot\theta = -\frac{12}{5}$.

1Step 1: Find the Radius r

Compute the radius $r$ using the Pythagorean theorem. The radius is given by the formula $r = \sqrt{x^{2} + y^{2}}$, where $x = -12$ and $y = 5$. Hence, $r = \sqrt{(-12)^{2} + 5^{2}} = \sqrt{144 + 25} = \sqrt{169} = 13$.

2Step 2: Calculate Sine

The sine function ($sin\theta$) is given by the ratio of the y-coordinate to the radius. So, $sin\theta = \frac{y}{r} = \frac{5}{13}$.

3Step 3: Calculate Cosine

The cosine function ($cos\theta$) is given by the ratio of the x-coordinate to the radius. So, $cos\theta = \frac{x}{r} = \frac{-12}{13}$.

4Step 4: Calculate Tangent

The tangent function ($tan\theta$) is given by the ratio of sine to cosine. So, $tan\theta = \frac{sin\theta}{cos\theta} = \frac{5/13}{-12/13} = -\frac{5}{12}$.

5Step 5: Calculate Cosecant

The cosecant function ($csc\theta$) is the reciprocal of sine. So, $csc\theta = \frac{1}{sin\theta} = \frac{13}{5}$.

6Step 6: Calculate Secant

The secant function ($sec\theta$) is the reciprocal of cosine. So, $sec\theta = \frac{1}{cos\theta} = -\frac{13}{12}$.

7Step 7: Calculate Cotangent

The cotangent function ($cot\theta$) is the reciprocal of tangent. So, $cot\theta = \frac{1}{tan\theta} = -\frac{12}{5}$.

Problem 2