Find the value of each of the six trigonometric functions of the angle 0 in the figure. a a = 10 and b=7 sin 0 = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
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Find the value of each of the 6 trig
![**Exercise: Trigonometric Functions Calculation**
*Objective*: Find the value of each of the six trigonometric functions for the angle \(\theta\) in the given right triangle.
**Given**:
- \(a = 10\)
- \(b = 7\)
**Instructions**:
1. Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.
2. Enter your answer in the answer box and then click 'Check Answer'.
**Diagrams and Detailed Steps**:
The diagram shows a right triangle with:
- The side opposite angle \(\theta\) labeled as \(a\).
- The side adjacent to angle \(\theta\) labeled as \(b\).
- The hypotenuse can be calculated.
To find the hypotenuse (\(c\)) of the triangle, use the Pythagorean theorem:
\[ c = \sqrt{a^2 + b^2} \]
Calculate \(c\):
\[ c = \sqrt{10^2 + 7^2} = \sqrt{100 + 49} = \sqrt{149} \]
Now, calculate each of the six trigonometric functions:
1. **Sine function (\(\sin \theta\))**:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{10}{\sqrt{149}} \]
2. **Cosine function (\(\cos \theta\))**:
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} = \frac{7}{\sqrt{149}} \]
3. **Tangent function (\(\tan \theta\))**:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{10}{7} \]
4. **Cosecant function (\(\csc \theta\))**:
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{149}}{10} \]
5. **Secant function (\(\sec \theta\))**:
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{149}}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2bbef39-13a9-4a3a-ae0f-bc83a28a95a1%2F445d1888-394b-453b-b32c-cd111757e1e3%2Fulk2njv.jpeg&w=3840&q=75)
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