Question

\(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Find the exact value of each expression if \(\theta=60^{\circ} .\) Do not use a calculator. $$ 2 f(\theta) $$

Short answer

\(2 f(60^{\circ}) = \sqrt{3}\)

Step-by-step solution

- Recognize the given functions

We are given the functions: \(f(\theta) = \sin \theta\) \(g(\theta) = \cos \theta\)and we need to find the value of \(2 f(\theta)\) when \(\theta = 60^{\circ} \).

- Substitute \(\theta = 60^{\circ}\) into \(f(\theta)\)

According to the given function \(f(\theta) = \sin \theta\), we substitute \(\theta\) with \(60^{\circ}\): \(f(60^{\circ}) = \sin 60^{\circ}\).

- Use the exact value of \(\sin 60^{\circ}\)

Recall the exact value: \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\).Hence, \(f(60^{\circ}) = \frac{\sqrt{3}}{2}\).

- Multiply by 2

We need to find \(2 f(\theta)\). So we multiply our result by 2:\(2 f(60^{\circ}) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}\).

Key concepts

Sine Function

The sine function, \( \sin\theta \), is one of the primary trigonometric functions. It is defined as the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse in a right-angled triangle.
The sine function is periodic with a period of \(2\pi\), meaning that its values repeat every \(2\pi\) radians.
In the unit circle, the sine of an angle \( \theta \) is the y-coordinate of the point where the terminal side of \( \theta \) intersects the circle.
Key properties of the sine function include:

• Range: \(-1 \leq \sin\theta \leq 1\)
• Symmetry: \( \sin(-\theta) = -\sin\theta\)
• Zeros: At multiples of \pi\, i.e., \(\theta = \pi, 2\pi, 3\pi, \ldots\)

Special Angles in Trigonometry

Certain angles in trigonometry, known as special angles, have exact trigonometric values. These angles are typically \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \text{and} \ 90^{\circ}\).
These special angles and their trigonometric values are often memorized because they frequently appear in math problems.
For example:

• \(\sin 30^{\circ} = \frac{1}{2}\)
• \(\text{and} \ \sin 45^{\circ} = \frac{\sqrt{2}}{2}\)
• \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)
• \(\sin 90^{\circ} = 1\)
  Knowing these values by heart makes solving trigonometric problems much quicker and easier.

Exact Trigonometric Values

Exact trigonometric values are values of trigonometric functions at special angles that are known to be precise.
They are not approximations but the true values. For example, \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), this value is not an approximation but an exact value.
Exact values are useful because they allow you to solve trigonometric problems without a calculator. Here are some exact values:

• \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
• \(\cos 45^{\circ} = \frac{1}{2}\)
• \(\cos 60^{\circ} = \frac{1}{2}\)
  By knowing these exact values, you can accurately solve trigonometric equations and expressions without needing to round or approximate.