Answer :
Sure, let's find the six trigonometric functions of the angle [tex]\( \theta \)[/tex] based on the point [tex]\((-5, -2)\)[/tex] on its terminal side.
### Step 1: Determine the Hypotenuse (r)
The hypotenuse, [tex]\( r \)[/tex], of a right triangle with legs corresponding to the coordinates [tex]\( x = -5 \)[/tex] and [tex]\( y = -2 \)[/tex] can be found using the Pythagorean theorem:
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
Substituting the given values:
[tex]\[ r = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.385 \][/tex]
### Step 2: Calculate the Sine Function
The sine function is defined as the ratio of the opposite side to the hypotenuse.
[tex]\[ \sin \theta = \frac{y}{r} = \frac{-2}{\sqrt{29}} \approx -0.371 \][/tex]
### Step 3: Calculate the Cosine Function
The cosine function is defined as the ratio of the adjacent side to the hypotenuse.
[tex]\[ \cos \theta = \frac{x}{r} = \frac{-5}{\sqrt{29}} \approx -0.928 \][/tex]
### Step 4: Calculate the Tangent Function
The tangent function is the ratio of the opposite side to the adjacent side.
[tex]\[ \tan \theta = \frac{y}{x} = \frac{-2}{-5} = 0.4 \][/tex]
### Step 5: Calculate the Cosecant Function
The cosecant function is the reciprocal of the sine function.
[tex]\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{2}{\sqrt{29}}} = \frac{\sqrt{29}}{-2} \approx -2.693 \][/tex]
### Step 6: Calculate the Secant Function
The secant function is the reciprocal of the cosine function.
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{5}{\sqrt{29}}} = \frac{\sqrt{29}}{-5} \approx -1.077 \][/tex]
### Step 7: Calculate the Cotangent Function
The cotangent function is the reciprocal of the tangent function.
[tex]\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{0.4} = 2.5 \][/tex]
### Summary of Trigonometric Functions:
- Hypotenuse (r): [tex]\( \approx 5.385 \)[/tex]
- Sine ([tex]\( \sin \theta \)[/tex]): [tex]\( \approx -0.371 \)[/tex]
- Cosine ([tex]\( \cos \theta \)[/tex]): [tex]\( \approx -0.928 \)[/tex]
- Tangent ([tex]\( \tan \theta \)[/tex]): [tex]\( 0.4 \)[/tex]
- Cosecant ([tex]\( \csc \theta \)[/tex]): [tex]\( \approx -2.693 \)[/tex]
- Secant ([tex]\( \sec \theta \)[/tex]): [tex]\( \approx -1.077 \)[/tex]
- Cotangent ([tex]\( \cot \theta \)[/tex]): [tex]\( 2.5 \)[/tex]
These are the exact trigonometric values for the angle [tex]\( \theta \)[/tex] where [tex]\((-5, -2)\)[/tex] is a point on its terminal side.
### Step 1: Determine the Hypotenuse (r)
The hypotenuse, [tex]\( r \)[/tex], of a right triangle with legs corresponding to the coordinates [tex]\( x = -5 \)[/tex] and [tex]\( y = -2 \)[/tex] can be found using the Pythagorean theorem:
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
Substituting the given values:
[tex]\[ r = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.385 \][/tex]
### Step 2: Calculate the Sine Function
The sine function is defined as the ratio of the opposite side to the hypotenuse.
[tex]\[ \sin \theta = \frac{y}{r} = \frac{-2}{\sqrt{29}} \approx -0.371 \][/tex]
### Step 3: Calculate the Cosine Function
The cosine function is defined as the ratio of the adjacent side to the hypotenuse.
[tex]\[ \cos \theta = \frac{x}{r} = \frac{-5}{\sqrt{29}} \approx -0.928 \][/tex]
### Step 4: Calculate the Tangent Function
The tangent function is the ratio of the opposite side to the adjacent side.
[tex]\[ \tan \theta = \frac{y}{x} = \frac{-2}{-5} = 0.4 \][/tex]
### Step 5: Calculate the Cosecant Function
The cosecant function is the reciprocal of the sine function.
[tex]\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{2}{\sqrt{29}}} = \frac{\sqrt{29}}{-2} \approx -2.693 \][/tex]
### Step 6: Calculate the Secant Function
The secant function is the reciprocal of the cosine function.
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{5}{\sqrt{29}}} = \frac{\sqrt{29}}{-5} \approx -1.077 \][/tex]
### Step 7: Calculate the Cotangent Function
The cotangent function is the reciprocal of the tangent function.
[tex]\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{0.4} = 2.5 \][/tex]
### Summary of Trigonometric Functions:
- Hypotenuse (r): [tex]\( \approx 5.385 \)[/tex]
- Sine ([tex]\( \sin \theta \)[/tex]): [tex]\( \approx -0.371 \)[/tex]
- Cosine ([tex]\( \cos \theta \)[/tex]): [tex]\( \approx -0.928 \)[/tex]
- Tangent ([tex]\( \tan \theta \)[/tex]): [tex]\( 0.4 \)[/tex]
- Cosecant ([tex]\( \csc \theta \)[/tex]): [tex]\( \approx -2.693 \)[/tex]
- Secant ([tex]\( \sec \theta \)[/tex]): [tex]\( \approx -1.077 \)[/tex]
- Cotangent ([tex]\( \cot \theta \)[/tex]): [tex]\( 2.5 \)[/tex]
These are the exact trigonometric values for the angle [tex]\( \theta \)[/tex] where [tex]\((-5, -2)\)[/tex] is a point on its terminal side.
