Question

Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume \(x>0.\) $$\sin ^{-1}(\cos \theta), \text { for } 0 \leq \theta \leq \pi / 2$$

Short answer

Question: Simplify the expression \(\sin^{-1}(\cos\theta)\) for an angle \(\theta\) between \(0\) and \(\pi/2\). Answer: \(\theta\) Explanation: Using a right triangle with angle \(\theta\) and the trigonometric relationships, we can find that \(\sin^{-1}(\cos\theta) = \theta\).

Step-by-step solution

Draw a right triangle

Draw a right triangle with one angle labeled as \(\theta\). Let the opposite side to \(\theta\) be \(a\), the adjacent side be \(b\), and the hypotenuse be \(c\).

Find the relationship between sides and the angle

Since we are given the expression \(\sin^{-1}(\cos\theta)\), we will first need to find an expression for \(\cos\theta\). Recall that: $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c}$$

Use Pythagorean theorem

To find the relationship between sides \(a\), \(b\), and \(c\), we can use the Pythagorean theorem: $$ a^2 + b^2 = c^2 $$

Find the values of sine and cosine

Here, we are given the cosine of the angle \(\theta\). Let's now find its sine. Recall that: $$ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} $$ Using the Pythagorean theorem, we can express \(a\) in terms of \(b\) and \(c\). Thus, $$ a^2 = c^2 - b^2$$ $$ a = \sqrt{c^2 - b^2}$$ Now, substituting \(a\) into the sine equation, we get $$ \sin\theta = \frac{\sqrt{c^2 - b^2}}{c} $$

Simplify the given expression

Now, we will simplify the expression \(\sin^{-1}(\cos\theta)\). Since \(\cos\theta = \frac{b}{c}\), we get $$ \sin^{-1}(\cos\theta) = \sin^{-1}\left(\frac{b}{c}\right)$$ Now recall that \(\sin\theta = \frac{\sqrt{c^2 - b^2}}{c}\). Since we want to find the angle \(\theta\), we need to invert the sine function: $$ \theta = \sin^{-1}\left(\frac{\sqrt{c^2 - b^2}}{c}\right) $$ By comparing this equation with our expression, we can see that $$ \sin^{-1}(\cos\theta) = \sin^{-1}\left(\frac{b}{c}\right) = \theta $$ Therefore, the expression simplifies to \(\theta\).

Key concepts

Right Triangle Essentials

A right triangle is a type of triangle with one angle measuring 90 degrees. This particular type of triangle is useful in trigonometry because it allows us to define the main trigonometric ratios: sine, cosine, and tangent. These ratios form the basis for many trigonometric identities and are based on the triangle's sides relative to its angles. In the context of solving expressions, we often visualize right triangles to simplify these calculations and understand relationships between angles and sides.

When solving problems involving trigonometric identities, it's beneficial to draw a right triangle:

• Label the angle of interest, often denoted as \( \theta \).
• Identify the sides: opposite, adjacent, and hypotenuse. For \( \theta \), the opposite side faces \( \theta \), the adjacent side is the base touching \( \theta \), and the hypotenuse is the longest side.

These visuals help translate complex expressions into more manageable geometric and algebraic terms.

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions, such as \( \sin^{-1}, \) are used to find angles when you know the trigonometric ratios. For example, if you know \( \cos \theta, \) you can use the inverse cosine function \( \cos^{-1} \) to find \( \theta. \) These functions are crucial for reversing trigonometric operations and solving for angles in triangles.

The inverse sine, \( \sin^{-1}, \) specifically handles inputs from the sine ratio to output an angle. For a given expression like \( \sin^{-1}(\cos \theta), \) you're essentially finding the angle whose sine would equal the cosine value. This approach is particularly handy in cases where you need to evaluate expressions involving various trigonometric identities.

Remember:

• Inverse functions are denoted with \( ^{-1} \).
• They output angles typically within specific ranges, like \( 0 \leq \theta \leq \frac{\pi}{2} \).

This ensures that every input provides a uniquely defined angle as output.

Exploring the Pythagorean Theorem

The Pythagorean theorem is a fundamental property of right triangles, expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. This relationship allows you to solve for any side of the triangle, given the other two.

In trigonometry, this theorem forms the backbone of understanding trigonometric ratios. For instance, to find sine and cosine values in a right triangle, you must determine the side lengths. Using the theorem, you can express one side in terms of the others and thus find:

• \( \cos \theta = \frac{b}{c} \)
• \( \sin \theta = \frac{a}{c} \)

The expression \( a = \sqrt{c^2 - b^2} \) comes from rearranging the Pythagorean theorem to solve for \( a \). This step illustrates how you can derive precise measurements required for understanding and manipulating trigonometric functions. Ultimately, applying this theorem simplifies complex trigonometric expressions by relating them directly to a triangle's geometry.