Math Problem Statement

Calculate sin (α/2)

Solution

To calculate sin(α2)\sin\left(\frac{\alpha}{2}\right) for the given right triangle, we'll first need to find the value of α\alpha, the angle at AA. Then, we'll use the half-angle identity to determine sin(α2)\sin\left(\frac{\alpha}{2}\right).

Step 1: Find α\alpha

In a right triangle, the sum of the angles is 9090^\circ (for the right angle at BB) plus the two acute angles α\alpha and θ\theta. Therefore: α=90θ\alpha = 90^\circ - \theta

From the previous calculation, we found tan(θ)=43\tan(\theta) = \frac{4}{3}, so: θ=arctan(43)\theta = \arctan\left(\frac{4}{3}\right)

Thus: α=90arctan(43)\alpha = 90^\circ - \arctan\left(\frac{4}{3}\right)

Step 2: Use the half-angle identity

The half-angle identity for sine is: sin(α2)=1cos(α)2\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \cos(\alpha)}{2}}

First, we need to find cos(α)\cos(\alpha). Using the identity cos(α)=sin(θ)\cos(\alpha) = \sin(\theta), and knowing: sin(θ)=oppositehypotenuse=810=0.8\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{10} = 0.8

Thus: cos(α)=0.8\cos(\alpha) = 0.8

Now, apply the half-angle identity: sin(α2)=10.82=0.22=0.1=1010\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - 0.8}{2}} = \sqrt{\frac{0.2}{2}} = \sqrt{0.1} = \frac{\sqrt{10}}{10}

So, sin(α2)=1010\sin\left(\frac{\alpha}{2}\right) = \frac{\sqrt{10}}{10}.

Would you like more details on any of these steps or do you have further questions?


Here are 8 related questions for further exploration:

  1. How do you calculate cos(α)\cos(\alpha) using the sides of the triangle?
  2. What is the exact value of tan(α2)\tan\left(\frac{\alpha}{2}\right)?
  3. How can you calculate cos(α2)\cos\left(\frac{\alpha}{2}\right)?
  4. What is the sine of angle θ\theta in this triangle?
  5. How would you calculate the length of the altitude from AA to side BCBC?
  6. What are the trigonometric identities for double angles α\alpha and θ\theta?
  7. How do you find the circumradius of triangle ABCABC?
  8. What is the relationship between sin(α)\sin(\alpha) and sin(θ)\sin(\theta) in this triangle?

Tip: When working with trigonometric identities, always double-check the values you obtain for angles to ensure that you are using the correct identity for the problem at hand.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle

Formulas

Half-angle identity for sine

Theorems

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Suitable Grade Level

Grades 10-12