Problem 65 - Precalculus : Building Concepts and Connections

Question

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. Let \(\alpha\) be an acute angle with \(\sin \alpha=a\). Find \(\csc \alpha\) and \(\cos \left(90^{\circ}-\alpha\right)\) in terms of \(a\)

Step-by-Step Solution

Answer

\(\csc \alpha = 1/a\) and \(\cos \left(90^{\circ}-\alpha\right) = a\)

Step 1: Define Cosecant

Cosecant is the reciprocal of sine. So, if we know that \(\sin \alpha=a\), we can directly find that \(\csc \alpha=\frac{1}{a}\)

Step 2: Apply Complementary Angle Theorem

The Complementary Angle Theorem states that the cosine of an angle is equivalent to the sine of its complement. It means that \(\cos \left(90^{\circ}-\alpha\right) equals to \(\sin \alpha\)

Step 3: Subsitute the given value

Finally we can substitute the known sine value into the equation from step 2. So, \(\cos \left(90^{\circ}-\alpha\right)= \sin \alpha= a\)

Key Concepts

Cosecant Function

The cosecant function is an integral part of trigonometry closely tied to the sine function. If you remember one thing, remember that cosecant is the reciprocal of sine.

• This means that if you have a sine value of an angle, denoted as \( \sin \alpha = a \), the cosecant is simply \( \csc \alpha = \frac{1}{\sin \alpha} = \frac{1}{a} \).
• This relationship helps in finding out the cosecant quickly when we know the sine.

Understanding this concept simplifies problem-solving in trigonometry, especially when dealing with right triangles and circle-based calculations.
The cosecant function is undefined for sine values of zero (since division by zero is undefined).
So, angle measurement must always consider this to avoid invalid mathematical operations.

Complementary Angle Theorem

The Complementary Angle Theorem is an essential tool in trigonometry that relates angles and their complements in right triangles. The complement of an angle is what it needs to add up to \( 90^{\circ} \), making them complementary.

• According to this theorem, for any angle \( \alpha \), the cosine of its complement (\(90^{\circ}-\alpha\)) is equal to the sine of the angle itself.
• This means \( \cos(90^{\circ}-\alpha) = \sin \alpha \).

This relationship is extremely useful, as it allows you to switch between sine and cosine functions with ease.
It means you don't always need separate values because knowing one immediately gives you a clue about the other.
This theorem is one of the basic identities used frequently and helps simplify calculations in many mathematical problems.

Acute Angle

In trigonometry, understanding different types of angles is crucial.
An acute angle is any angle less than \(90^{\circ}\). It is a common angle type used in triangle calculations, especially right triangles.

• An acute angle is present in all non-right angles of a right triangle, ensuring their combined value with \(90^{\circ}\) never surpasses \(180^{\circ}\).
• Trigonometric functions like sine, cosine, or tangent will always subject positively for acute angles, as it lies within the first quadrant of the unit circle.

The acute angle plays a central role in solving trigonometric problems due to its predictable properties.
Whether it involves direct calculations or deciphering an angle's implications in a geometric context, understanding the concept of acute angles helps ensure accurate and effective approaches in trigonometry.