Problem 18 - Algebra 2

Question

Mental Math Find the value of each trigonometric expression. $$ \sin 80^{\circ} \cos 35^{\circ}-\cos 80^{\circ} \sin 35^{\circ} $$

Step-by-Step Solution

Answer

The value of the trigonometric expression is \(\frac{\sqrt{2}}{2}\).

Step 1: Identify and apply the appropriate trigonometric identity

Looking at the exercise, it can be observed that it fits the mold of the subtractive identity for sine which is \(\sin(A - B) = \sin A \cos B - \cos A \sin B\) . Therefore, the given expression \(\sin 80^{\circ} \cos 35^{\circ} - \cos 80^{\circ} \sin 35^{\circ}\) can be rewritten using this identity as \(\sin(80^{\circ}-35^{\circ})\).

Step 2: Calculate the value of the expression

With the simplification in step 1, the expression now becomes \(\sin(80^{\circ}-35^{\circ})\). Therefore, calculate the value of \(80^{\circ}-35^{\circ}\) which is \(45^{\circ}\). The sine of \(45^{\circ}\) is \(\frac{\sqrt{2}}{2}\), which is the solution.

Key Concepts

Sine Subtraction Identity

The Sine Subtraction Identity is a fundamental concept in trigonometry. It helps simplify trigonometric expressions involving the sine function. In particular, it allows you to express the sine of the difference of two angles in a simpler form.
For the Sine Subtraction Identity, the formula is:

• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

This means that if you have an expression like \( \sin A \cos B - \cos A \sin B \), you can simplify it by rewriting it as \( \sin(A - B) \).
This identity is immensely useful for solving trigonometric equations, proving other identities, and simplifying complex expressions. Understanding and applying this identity allows you to solve problems more efficiently.
In our exercise, we used this identity to transform \( \sin 80^{\circ} \cos 35^{\circ} - \cos 80^{\circ} \sin 35^{\circ} \) into \( \sin(80^{\circ} - 35^{\circ}) \), making the problem much easier to solve.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are essential in many areas of mathematics and science, including geometry, physics, and engineering.
The most common trigonometric functions include:

• Sine (\( \sin \))
• Cosine (\( \cos \))
• Tangent (\( \tan \))

These functions are defined based on a right triangle and its ratios. For instance, the sine of an angle is the ratio of the opposite side to the hypotenuse.
Trigonometric functions are periodic, meaning they repeat their values in regular intervals. This characteristic is vital in modeling cyclical phenomena, like waves.
In our problem, understanding the sine and cosine functions was crucial for applying the Sine Subtraction Identity and solving the expression effectively.

Mental Math in Trigonometry

Mental math skills can greatly enhance the efficiency of solving trigonometric problems. While calculators and software can solve trigonometric expressions, being able to compute mentally aids in quick estimation and verification.
When dealing with basic angles, like \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \), memorizing their sine, cosine, and tangent values is extremely helpful. For instance:

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

By understanding these basic trigonometric values, you can solve problems more swiftly without relying on external aids.
In the exercise, after simplifying the expression to \( \sin(45^{\circ}) \), we used mental math to know that \( \sin 45^{\circ} \) equals \( \frac{\sqrt{2}}{2} \). This kind of recognition makes solving such trigonometric problems much more straightforward and efficient.