Math Problem Statement

Find cot(θ), csc(θ), and sin(θ) where θ is the angle shown in the figure, using exact values.

Solution

To solve the given problem, let's analyze the triangle and find the values of cotθ\cot \theta, cscθ\csc \theta, and sinθ\sin \theta using trigonometric identities.

Step 1: Identify the sides of the triangle

The right triangle shows:

  • Opposite side to θ\theta: 1515
  • Adjacent side to θ\theta: Unknown for now
  • Hypotenuse: 1717

We can find the adjacent side using the Pythagorean theorem: Adjacent side=172152\text{Adjacent side} = \sqrt{17^2 - 15^2} Let's calculate it: 172=289,152=22517^2 = 289, \quad 15^2 = 225 289225=64289 - 225 = 64 Adjacent side=64=8\text{Adjacent side} = \sqrt{64} = 8

Now, we have:

  • Opposite side: 1515
  • Adjacent side: 88
  • Hypotenuse: 1717

Step 2: Find cotθ\cot \theta, cscθ\csc \theta, and sinθ\sin \theta

  1. sinθ=oppositehypotenuse=1517\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{15}{17}
  2. cscθ=1sinθ=1715\csc \theta = \frac{1}{\sin \theta} = \frac{17}{15}
  3. cotθ=adjacentopposite=815\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{8}{15}

Final answers:

  • cotθ=815\cot \theta = \frac{8}{15}
  • cscθ=1715\csc \theta = \frac{17}{15}
  • sinθ=1517\sin \theta = \frac{15}{17}

Would you like a more detailed explanation or have any questions?

Related Questions:

  1. How do you find the remaining trigonometric ratios (cosθ\cos \theta, tanθ\tan \theta, secθ\sec \theta) for this triangle?
  2. What is the value of θ\theta in degrees or radians?
  3. How does the Pythagorean identity relate to this triangle?
  4. How can you determine θ\theta if given only cotθ\cot \theta?
  5. Why is cscθ\csc \theta the reciprocal of sinθ\sin \theta?

Tip:

In right triangles, always start with the Pythagorean theorem to find missing sides, which helps in determining all trigonometric ratios accurately.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles

Formulas

sin(θ) = opposite / hypotenuse
csc(θ) = 1 / sin(θ)
cot(θ) = adjacent / opposite
Pythagorean Theorem: a^2 + b^2 = c^2

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 9-12