Problem 25 - Algebra and Trigonometry

Question

Use an identity to find the value of each expression. Do not use a calculator. $$ \sin 37^{\circ} \mathrm{csc} 37^{\circ} $$

Step-by-Step Solution

Answer

The value of the expression \( \sin 37^{\circ} csc 37^{\circ} \) is 1.

Step 1: Apply the Identity for Cosecant

First, replace \(csc 37^{\circ}\) with its equivalent expression based on the identity for cosecant, that is, \(cscA = \frac{1}{sinA}\). So we get: \( \sin 37^{\circ} *\frac{1}{\sin 37^{\circ}}\)

Step 2: Simplify the Expression

In the resulting expression \( \sin 37^{\circ} * \frac{1}{\sin 37^{\circ}}\), note that the same term appears in the numerator and the denominator. So, they cancel each other out, leaving only 1 as the resulting value.

Key Concepts

Cosecant Identity

The cosecant identity is a fundamental trigonometric principle that simplifies many mathematical expressions. It relates cosecant to sine, one of the primary trigonometric functions.
The cosecant of an angle is defined as the reciprocal of the sine function of that angle. Thus, the identity is written as:

• \( ext{csc} A = \frac{1}{ ext{sin} A}\)

This relationship helps simplify expressions involving both sine and cosecant.
In practical terms, if you have a trigonometric expression that includes a sine and a cosecant term of the same angle, you can replace the cosecant with \(\frac{1}{\text{sin}A}\). This substitution often reveals further simplifications, as it did in our original problem where \(\sin 37^{\circ} \times \text{csc} 37^{\circ}\) becomes \(\sin 37^{\circ} \times \frac{1}{\sin 37^{\circ}}\), ultimately simplifying to 1.

Sine Function

The sine function is a fundamental building block in trigonometry and represents a relationship in a right triangle. Sine is usually abbreviated as 'sin', and it expresses the ratio of the length of the side opposite the angle to the length of the hypotenuse.
For any angle \(A\), it is represented as:

• \( ext{sin} A = \frac{ ext{Opposite}}{ ext{Hypotenuse}}\)

The sine function is crucial in various applications, including physics, engineering, and geometry.
It also plays a central role in trigonometric identities, like the one seen in our exercise, where combining the sine and its reciprocal, the cosecant, leads to simplification.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra and trigonometry, helping to make calculations more manageable.
When you see repeated terms like in \(\sin 37^{\circ} \times \frac{1}{\sin 37^{\circ}}\), it's a clear opportunity to simplify by canceling.

• In this expression, both the sine term and its reciprocal are present.
• The mathematical multiplication between a term and its reciprocal equals 1.

Thus, replacing \(\text{csc} 37^{\circ}\) with \(\frac{1}{\sin 37^{\circ}}\) allowed us to narrowly focus on the nature of recant expressions.
Simplification, as demonstrated, converts complex calculations into straightforward results, promoting more efficient problem-solving skills.