Math Problem Statement

Sketch a right triangle corresponding to the trigonometric function of the acute angle πœƒ. Then find the exact values of the other five trigonometric functions of πœƒ. sin(πœƒ) = 5 13 cos(πœƒ) =

tan(πœƒ) =

csc(πœƒ) =

sec(πœƒ) =

cot(πœƒ) =

Solution

We are given that sin⁑(θ)=513\sin(\theta) = \frac{5}{13}, and we need to find the remaining five trigonometric functions of the acute angle θ\theta. Let's proceed step by step.

Step 1: Set up a right triangle based on sin⁑(θ)\sin(\theta)

The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. So we can say:

  • Opposite side =5= 5
  • Hypotenuse =13= 13

Now, we need to find the adjacent side using the Pythagorean theorem: (Hypotenuse)2=(Opposite)2+(Adjacent)2\text{(Hypotenuse)}^2 = (\text{Opposite})^2 + (\text{Adjacent})^2 132=52+(Adjacent)213^2 = 5^2 + (\text{Adjacent})^2 169=25+(Adjacent)2169 = 25 + (\text{Adjacent})^2 (Adjacent)2=144(\text{Adjacent})^2 = 144 Adjacent=144=12\text{Adjacent} = \sqrt{144} = 12

Step 2: Find the remaining trigonometric functions

Now that we know the opposite side is 5, the adjacent side is 12, and the hypotenuse is 13, we can find the other trigonometric functions:

  1. Cosine: cos⁑(θ)=AdjacentHypotenuse=1213\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{12}{13}

  2. Tangent: tan⁑(θ)=OppositeAdjacent=512\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{12}

  3. Cosecant: csc⁑(θ)=1sin⁑(θ)=135\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{13}{5}

  4. Secant: sec⁑(θ)=1cos⁑(θ)=1312\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{13}{12}

  5. Cotangent: cot⁑(θ)=1tan⁑(θ)=125\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{12}{5}

Summary of all trigonometric functions for ΞΈ\theta:

  • sin⁑(ΞΈ)=513\sin(\theta) = \frac{5}{13}
  • cos⁑(ΞΈ)=1213\cos(\theta) = \frac{12}{13}
  • tan⁑(ΞΈ)=512\tan(\theta) = \frac{5}{12}
  • csc⁑(ΞΈ)=135\csc(\theta) = \frac{13}{5}
  • sec⁑(ΞΈ)=1312\sec(\theta) = \frac{13}{12}
  • cot⁑(ΞΈ)=125\cot(\theta) = \frac{12}{5}

If you would like, I can create a sketch of this triangle for better visualization.

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


5 related questions:

  1. How does the Pythagorean theorem relate to trigonometric functions?
  2. What are the properties of the reciprocal trigonometric functions?
  3. How do you find trigonometric functions for angles in non-right triangles?
  4. What happens to the trigonometric functions if the angle ΞΈ\theta is in a different quadrant?
  5. How can trigonometric identities be used to simplify expressions involving these functions?

Tip: Remember that the reciprocal trigonometric functions (csc, sec, cot) can be found directly from their primary counterparts (sin, cos, tan).

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles
Trigonometric Ratios

Formulas

sin(πœƒ) = opposite/hypotenuse
cos(πœƒ) = adjacent/hypotenuse
tan(πœƒ) = opposite/adjacent
csc(πœƒ) = 1/sin(πœƒ)
sec(πœƒ) = 1/cos(πœƒ)
cot(πœƒ) = 1/tan(πœƒ)
Pythagorean theorem: a^2 + b^2 = c^2

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 9-12