Math Problem Statement
Solution
The given image shows a right triangle, where the hypotenuse is , the adjacent side to angle is 2, and the opposite side can be determined using the Pythagorean theorem. Let’s calculate the six trigonometric functions for .
Step 1: Use the Pythagorean Theorem
The formula is: Substituting the values:
Now the triangle's sides are:
- Adjacent:
- Opposite:
- Hypotenuse:
Step 2: Trigonometric Functions
-
Sine ():
-
Cosine ():
-
Tangent ():
-
Cosecant ():
-
Secant ():
-
Cotangent ():
Final Answer:
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Related Questions:
- How can you find trigonometric functions for a given angle in any triangle?
- What is the significance of rationalizing fractions in trigonometry?
- How are the reciprocal functions (e.g., secant, cosecant) related to sine and cosine?
- How can you use trigonometric ratios to solve for missing angles in a triangle?
- What are some real-world applications of trigonometry?
Tip:
Always verify the triangle’s side lengths using the Pythagorean theorem to avoid errors in trigonometric calculations.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangle
Trigonometric Ratios
Formulas
Pythagorean Theorem: a² + b² = c²
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent
csc(θ) = 1 / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1 / tan(θ)
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 9-12
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