Problem 24 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \cos \left(\operatorname{Arcsin} \frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The value is 0.80.

Step 1: Understand the Problem

We need to find the cosine of the angle whose sine is \( \frac{3}{5} \). This involves using trigonometric identities to relate sine and cosine.

Step 2: Set Up the Right Triangle

Consider a right triangle where the angle \( \theta \) has \( \sin(\theta) = \frac{3}{5} \). This means the opposite side is 3 and the hypotenuse is 5.

Step 3: Find the Adjacent Side

Use the Pythagorean theorem to find the adjacent side: \( a^2 + 3^2 = 5^2 \), simplifying to \( a^2 = 25 - 9 = 16 \), so \( a = 4 \).

Step 4: Determine \( \cos(\theta) \)

Now that we have the adjacent side (4) and the hypotenuse (5), use the definition of cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \).

Step 5: Verify the Result

We have used correct trigonometric identities and calculations, therefore \( \cos(\operatorname{Arcsin}\frac{3}{5}) = \frac{4}{5} \). Finally, express it in decimal form as 0.80.

Key Concepts

Right Triangle

A right triangle is a triangle in which one angle is exactly 90 degrees. This angle, known as the right angle, is fundamental to understanding trigonometric properties and calculations. There are three sides of a right triangle:

• The hypotenuse: the side opposite the right angle, and it is the longest side.
• The opposite side: the side opposite the angle of interest.
• The adjacent side: the side next to the angle of interest, excluding the hypotenuse.

When working with right triangles, these side labels help in defining and calculating trigonometric ratios like sine, cosine, and tangent. In the context of our exercise, knowing that the sine of an angle is given allows us to determine other properties of the triangle by leveraging the fundamental relationships between its sides.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that defines the relationship between the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (\( c \)) is equal to the sum of the squares of the other two sides (\( a \) and \( b \)). This can be represented by the formula:\[a^2 + b^2 = c^2\]In the problem, the hypotenuse is given as 5, and one of the other sides, the opposite, is 3. By applying the theorem, \(3^2 + a^2 = 5^2\), it quickly simplifies to \(a^2 = 16\). Taking the square root, \(a = 4\). This step is crucial to finding the adjacent side, which then allows you to determine other trigonometric functions like cosine.

Sine and Cosine Relationship

Sine and cosine are the foundational trigonometric functions used to relate the angles of a triangle to the lengths of its sides. The sine (\( \sin \)) of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse, while cosine (\( \cos \)) is the ratio of the adjacent side to the hypotenuse.
In this particular exercise, sine is given as \(\frac{3}{5}\). After determining the adjacent side using the Pythagorean theorem, the cosine can be calculated. Using the definition \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\), and given \(\text{adjacent} = 4\) and \(\text{hypotenuse} = 5\), we find \(\cos(\theta) = \frac{4}{5}\). This complementary relationship between sine and cosine is often used to solve trigonometric problems effectively.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle when we know the ratio of the sides. For instance, the arcsin \((\operatorname{Arcsin})\) function returns the angle whose sine is a specific value. This is essential in solving problems where you have a side ratio and need to determine the corresponding angle.
In our case, \(\operatorname{Arcsin}(\frac{3}{5})\) represents the angle \(\theta\) such that \(\sin(\theta) = \frac{3}{5}\). Once \(\theta\) is known, other trigonometric functions, like cosine, can be found using this angle. The angle in radians is especially useful when performing calculations that involve circle properties and periodic functions. Using inverse trigonometric functions widens our ability to tackle complex trigonometric equations by translating known side ratios back into angles.