Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\sin \left(\arctan \frac{4}{3}\right)\)

Step-by-Step Solution

Answer

The exact value of the expression \( \sin \left( \arctan \frac{4}{3} \right) \) is \( \frac{4}{5} \).

Step 1: Interpret the arctan function in terms of a right triangle

In a right triangle, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. When it is given that \( \arctan \left( \frac{4}{3}\right) \), it implies a right triangle where the length of the opposite side is 4 and the adjacent side is 3. From this, we can calculate the hypotenuse using the Pythagorean Theorem.

Step 2: Compute the Hypotenuse

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, the equation would be: \( \sqrt{4^2 + 3^2} \). Calculating the hypotenuse gives a value of 5.

Step 3: Determine the sine of the angle

Now that we have all three sides of the triangle, we can find the sine of the angle. Recall that the sine of an angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse. Based on the triangle we've outlined, computing the sine gives: \( \sin \left(\arctan \frac{4}{3} \right) = \frac{4}{5} \).

Step 4: Verify with a graphing utility

As a final step, the result can be checked with a graphing utility. Calculate the sine of the arctangent of 4/3. The graphing utility should also show a result of \( \frac{4}{5} \), verifying the initial answer.

Key Concepts

Right Triangle

Understanding the concept of a right triangle is the first crucial step in solving trigonometric problems using identities. A right triangle is a type of triangle that features one angle exactly equal to 90 degrees. This angle is called the right angle, and it is often denoted by a little square in diagram representations. The other two angles are always acute, meaning they are less than 90 degrees.

Each side of a right triangle has a unique relationship with the angles and are therefore named accordingly:

• Opposite side: The side opposite the angle we are focusing on, not the right angle.
• Adjacent side: The side right next to the angle considered, not the hypotenuse.
• Hypotenuse: The longest side opposite the right angle.

Understanding these sides is essential when working with trigonometric identities. They relate directly to ratios such as sine, cosine, and tangent, which are foundational in trigonometry.

Arctan Function

The arctan function, also known as the inverse tangent function, plays an important role in trigonometry. It is used to find an angle whose tangent ratio is a given value. When we talk about \( \arctan \left(\frac{4}{3}\right)\), it refers to an angle \( \theta \) such that the tangent of \( \theta \) is \( \frac{4}{3}\).

This is particularly important when dealing with right triangles because:

• The numerator 4 represents the opposite side of the right triangle.
• The denominator 3 indicates the adjacent side.

By reconstructing the triangle using these sides, we can leverage inverse trigonometric functions to find the unknown angle, linking back to how trigonometric ratios govern the sides and angles of a right triangle.

Pythagorean Theorem

The Pythagorean Theorem is a powerful tool in trigonometry and geometry. It allows us to determine the length of the hypotenuse in a right triangle when the lengths of the other two sides are known. The theorem states: \[ a^2 + b^2 = c^2 \]where \( a \) and \( b \) are the lengths of the two shorter sides known as the legs, and \( c \) is the hypotenuse.

In solving problems like \( \sin \left(\arctan \frac{4}{3}\right) \), this theorem becomes essential. With our example:

• \( a = 4 \) (opposite side)
• \( b = 3 \) (adjacent side)

Using the Pythagorean Theorem, we get:\[ c = \sqrt{4^2 + 3^2} = 5 \]By confirming the hypotenuse, we can further explore other trigonometric functions like sine and cosine, deriving various properties based on the side ratios.