Question

Draw a right triangle to simplify the given expressions. $$\sin ^{-1}(\cos \theta)$$

Short answer

Answer: \(90^{\circ} - \theta\)

Step-by-step solution

Draw a right triangle with angle θ

Draw a right triangle with one angle marked as \(\theta\). Label the sides of the triangle as follows: - Opposite side (the side opposite the angle \(\theta\)): O - Adjacent side (the side next to the angle \(\theta\) but not the hypotenuse): A - Hypotenuse: H

Express cosine θ in terms of the triangle sides

By definition, the cosine of an angle θ is equal to the ratio of the adjacent side to the hypotenuse. In our right triangle, this can be written as: $$\cos \theta = \frac{A}{H}$$

Think of another angle in the triangle

The other non-right angle in the triangle is \(90^{\circ} - \theta\). We want to find the sine of this angle in terms of the sides of the triangle, as it will help us simplify the expression.

Express sine(90° - θ) in terms of the triangle sides

By definition, the sine of an angle is equal to the ratio of the opposite side to the hypotenuse. In our right triangle, the sine of the angle \((90^{\circ} - \theta)\) can be written as: $$\sin(90^{\circ} - \theta) = \frac{A}{H}$$

Simplify the given expression

We know that the cosine of angle θ is equal to the ratio of the adjacent side to the hypotenuse: $$\cos \theta = \frac{A}{H}$$ And we also know that the sine of the angle \((90^{\circ} - \theta)\) is equal to the same ratio: $$\sin(90^{\circ} - \theta) = \frac{A}{H}$$ Since the given expression is \(\sin^{-1}(\cos \theta)\), we can now substitute the angle \((90^{\circ} - \theta)\) as the equivalent angle for the inverse sine function: $$\sin^{-1}(\cos \theta) = 90^{\circ} - \theta$$ So, the simplified expression for \(\sin^{-1}(\cos \theta)\) is: $$90^{\circ} - \theta$$

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle exactly equal to 90 degrees. This special property creates unique relationships between the sides and angles of the triangle. In a right triangle, there are three sides:

• Hypotenuse: The longest side, opposite the right angle.
• Adjacent Side: The side that forms the angle in question, alongside the hypotenuse.
• Opposite Side: The side opposite to the angle in question.

The right triangle is crucial for understanding the basic trigonometric functions. By visualizing a right triangle, you can better grasp how these functions work and how they are applied in various mathematical problems.

Cosine Function

The cosine function is one of the primary trigonometric functions, often abbreviated as 'cos'. It relates the angle of a right triangle to the ratio of its adjacent side to its hypotenuse. Given angle \(\theta\) in a right triangle, the cosine is expressed as:

• \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)

This function is particularly useful in determining the side lengths and angles of triangles. It also plays a fundamental role in Fourier transforms and oscillatory functions, appearing throughout physics and engineering. Its periodic nature makes it pivotal for exploring wave-like phenomena.

Sine Function

The sine function, abbreviated as 'sin', is another essential trigonometric function. For a given angle \(\theta\) in a right triangle, it relates this angle to the ratio of the opposite side to the hypotenuse:

• \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)

Like the cosine function, the sine function is also periodic and oscillatory. It is commonly used to model periodic phenomena like sound waves and light waves. In addition, the sine function is integral to the unit circle definition of trigonometric functions, extending beyond 0 to 90 degrees to a full 360-degree cycle.

Angle Subtraction Identity

The angle subtraction identity is a powerful trigonometric formula that expresses trigonometric functions of angles in terms of other angles. One famous identity involving both sine and cosine is:

• \(\sin(90^{\circ} - \theta) = \cos \theta\)

This identity signifies the complementary relationship between the sine and cosine functions. In the context of a right triangle, it essentially states that the sine of an angle is equal to the cosine of its complementary angle (i.e., 90 degrees minus the angle). This insight is aligned with how triangles behave under rotation and is used in deriving other trigonometric properties and simplifying expressions.