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Let's solve the problem by determining the six trigonometric functions of the given angle \( \theta \), based on the point \( (-10, 24) \) on the terminal side of the angle.

### Step 1: Determine the radius (hypotenuse) \( r \)
The radius \( r \) is given by the Pythagorean theorem:
\[
r = \sqrt{x^2 + y^2}
\]
Substituting \( x = -10 \) and \( y = 24 \):
\[
r = \sqrt{(-10)^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26
\]
So, \( r = 26 \).

### Step 2: Find the six trigonometric functions
The six trigonometric functions are based on the definitions:
1. \( \sin \theta = \frac{y}{r} \)
2. \( \cos \theta = \frac{x}{r} \)
3. \( \tan \theta = \frac{y}{x} \)
4. \( \csc \theta = \frac{r}{y} \) (reciprocal of sine)
5. \( \sec \theta = \frac{r}{x} \) (reciprocal of cosine)
6. \( \cot \theta = \frac{x}{y} \) (reciprocal of tangent)

#### 1. \( \sin \theta \):
\[
\sin \theta = \frac{24}{26} = \frac{12}{13}
\]

#### 2. \( \cos \theta \):
\[
\cos \theta = \frac{-10}{26} = \frac{-5}{13}
\]

#### 3. \( \tan \theta \):
\[
\tan \theta = \frac{24}{-10} = -\frac{12}{5}
\]

#### 4. \( \csc \theta \) (reciprocal of \( \sin \theta \)):
\[
\csc \theta = \frac{26}{24} = \frac{13}{12}
\]

#### 5. \( \sec \theta \) (reciprocal of \( \cos \theta \)):
\[
\sec \theta = \frac{26}{-10} = -\frac{13}{5}
\]

#### 6. \( \cot \theta \) (reciprocal of \( \tan \theta \)):
\[
\cot \theta = \frac{-10}{24} = -\frac{5}{12}
\]

### Final Answers:
1. \( \sin \theta = \frac{12}{13} \)
2. \( \cos \theta = \frac{-5}{13} \)
3. \( \tan \theta = -\frac{12}{5} \)
4. \( \csc \theta = \frac{13}{12} \)
5. \( \sec \theta = -\frac{13}{5} \)
6. \( \cot \theta = -\frac{5}{12} \)

Would you like further details on any of these steps or have any questions?

### Related Questions:
1. What quadrant is the angle \( \theta \) located in, based on the coordinates \( (-10, 24) \)?
2. How would the values of the trigonometric functions change if the point were in a different quadrant?
3. Can you use the same approach for an angle in standard position with different coordinates?
4. What are the reference angles for the given values of \( \sin \theta \) and \( \cos \theta \)?
5. How do the signs of trigonometric functions differ in different quadrants?

### Tip:
Always remember that the signs of the trigonometric functions depend on the quadrant in which the terminal side of the angle lies.

The point given below is on the terminal side of an angle θ. Find the exact value of each of the six trigonometric functions of θ. (-10, 24)

Explore how to determine the six trigonometric functions sin, cos, tan, csc, sec, and cot for an angle whose terminal side passes through the point (-10, 24). This detailed guide includes steps using the Pythagorean theorem and trigonometric identities, ideal for students in Grades 10-12.

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