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Chapter 5: Problem 96

Give an example of an angle \(\theta\) such that both \(\sin \theta\) and \(\sin (2 \theta)\) are rational.

Short Answer

Expert verified
An example of an angle 胃 such that both \(\sin \theta\) and \(\sin (2\theta)\) are rational is 胃 = 30掳, as \(\sin(30掳) = \frac{1}{2}\) and \(\sin(2 \times 30掳) = \sin(60掳) = \frac{\sqrt{3}}{2}\).

Step by step solution

01

Define the problem

First, we need to find an angle 胃 which satisfies both conditions, sin(胃) must be a rational number and sin(2胃) should also be a rational number.
02

Recognize sin(2胃) is a trigonometric identity

We should recognize that sin(2胃) is a trigonometric identity, which can alternatively be expressed as: sin(2胃) = 2sin(胃)cos(胃)
03

Find a right triangle with a rational sin(胃)

Since sine and cosine are also trigonometric ratios, we can first try to find a right triangle that has a rational value of sin(胃). We can start by considering the special 45-45-90 or 30-60-90 triangles. For the 30-60-90 triangle: the angles are 30掳, 60掳, and 90掳, and its sides are in the ratio 1:鈭�3:2. Sin(30掳) = opposite side / hypotenuse = 1/2, a rational number We can consider 胃 = 30掳.
04

Calculate sin(2胃) using the angle 胃 obtained in Step 3

Now that we have obtained an angle 胃 (30掳) with a rational sin(胃), we will check if sin(2胃) is also a rational number. Using the angle from Step 3: 胃 = 30掳 2胃 = 2(30掳) = 60掳 Now, we calculate sin(60掳) using the same 30-60-90 triangle: sin(60掳) = opposite side / hypotenuse = 鈭�3/2 Sin(60掳) = 鈭�3/2, which is also a rational number since it can be expressed as a fraction.
05

Provide the example angle 胃

We have found an angle 胃 = 30掳 where both sin(胃) and sin(2胃) are rational numbers: sin(30掳) = 1/2 sin(2 脳 30掳) = sin(60掳) = 鈭�3/2. Therefore, 胃 = 30掳 is an example of an angle such that both sin(胃) and sin(2胃) are rational numbers.

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