To find the values of all six trigonometric functions for an angle β whose terminal side goes through the point $(13, -15)$, we need to follow these steps:

Step 1: Find the Hypotenuse (Radius)

The hypotenuse (or radius, $r$) is calculated using the distance formula: $r = \sqrt{x^2 + y^2}$ where $x = 13$ and $y = -15$.

$r = \sqrt{13^2 + (-15)^2} = \sqrt{169 + 225} = \sqrt{394} \approx 19.8$

Step 2: Define the Trigonometric Functions

Using $x = 13$, $y = -15$, and $r \approx 19.8$:

1. Sine $\sin(\beta) = \frac{y}{r}$: $\sin(\beta) = \frac{-15}{19.8} \approx -0.8$

2. Cosine $\cos(\beta) = \frac{x}{r}$: $\cos(\beta) = \frac{13}{19.8} \approx 0.7$

3. Tangent $\tan(\beta) = \frac{y}{x}$: $\tan(\beta) = \frac{-15}{13} \approx -1.2$

4. Cosecant $\csc(\beta) = \frac{r}{y}$: $\csc(\beta) = \frac{19.8}{-15} \approx -1.3$

5. Secant $\sec(\beta) = \frac{r}{x}$: $\sec(\beta) = \frac{19.8}{13} \approx 1.5$

6. Cotangent $\cot(\beta) = \frac{x}{y}$: $\cot(\beta) = \frac{13}{-15} \approx -0.9$

Summary of Correct Values

• $\sin(\beta) \approx -0.8$
• $\cos(\beta) \approx 0.7$
• $\tan(\beta) \approx -1.2$
• $\csc(\beta) \approx -1.3$
• $\sec(\beta) \approx 1.5$
• $\cot(\beta) \approx -0.9$

From the image, it appears that the errors are in the secant and cotangent values. The correct values should be:

• sec($\beta$) ≈ 1.5
• cot($\beta$) ≈ -0.9

Would you like more details on any part of this solution?

Related Questions

1. How do we determine the quadrant in which angle β lies?
2. Why is it important to keep track of signs in trigonometric functions?
3. What are reciprocal trigonometric identities, and how are they applied here?
4. How would the trigonometric values change if the point was in a different quadrant?
5. What is the significance of rounding answers to one decimal place?

Tip

Always check if the angle's terminal point lies in a specific quadrant to determine the correct signs for trigonometric functions.