Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use right triangles to evaluate the expression. $$\sin \left(\cos ^{-1} \frac{4}{5}+\sin ^{-1} \frac{3}{5}\right)$$

Step-by-Step Solution

Answer

The evaluated expression \(\sin \left(\cos ^{-1} \frac{4}{5}+\sin ^{-1} \frac{3}{5}\right)\) equals \(\frac{24}{25}\).

Step 1 - Find the Missing Sides of the Triangles

We begin by drawing two right triangles. The first triangle corresponds to the angle whose cosine equals \(\frac{4}{5}\), and the second triangle relates to the angle whose sine equals \(\frac{3}{5}\). For the first triangle, let us mark the adjacent side as 4 (matching the numerator of the given fraction), the hypotenuse as 5 (matching the denominator of the given fraction), and use the Pythagorean theorem to compute the opposite side: \(\sqrt{5^2-4^2}=\sqrt{9}=3\). For the second triangle, we mark the opposite side as 3, the hypotenuse as 5, and again use the Pythagorean theorem to establish the adjacent side: \(\sqrt{5^2-3^2}=\sqrt{16}=4\).

Step 2 - Apply the Additive Formula for Sine

First, recall the additive formula for sine: \(\sin(a+b)=\sin a \cos b + \cos a \sin b\). We substitute \(a=\cos^{-1} \frac{4}{5}\) and \(b=\sin^{-1} \frac{3}{5}\). Then we apply the Pythagorean identities \(\sin(\cos^{-1} x)=\sqrt{1-x^2}\) and \(\cos(\sin^{-1} x)=\sqrt{1-x^2}\) to obtain,\(\sin(a+b)=\sin a \cos b + \cos a \sin b=\sin(\cos^{-1} \frac{4}{5}) \cos(\sin^{-1} \frac{3}{5}) + \cos(\cos^{-1} \frac{4}{5}) \sin(\sin^{-1} \frac{3}{5})=\sqrt{1-\left(\frac{4}{5}\right)^2} \sqrt{1-\left(\frac{3}{5}\right)^2}+ \frac{4}{5} \times \frac{3}{5}\)

Step 3 - Simplify the Expression

Carrying on from Step 2, we have \(\sin(\cos^{-1} \frac{4}{5}+\sin^{-1} \frac{3}{5})=\sqrt{1-\left(\frac{4}{5}\right)^2} \sqrt{1-\left(\frac{3}{5}\right)^2}+\frac{4}{5} \times \frac{3}{5}\), which simplifies to \(\sqrt{\left(\frac{9}{25}\right)}\times \sqrt{\left(\frac{16}{25}\right)}+\frac{12}{25}\). Further simplification gives us \(\frac{3}{5} \times \frac{4}{5} + \frac{12}{25} = \frac{12}{25} + \frac{12}{25} = \frac{24}{25}\)

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry, particularly useful when dealing with angles and their relationships in right triangles. They allow us to simplify complex trigonometric expressions or to prove that one side of an equation equals another.
A few key trigonometric identities are often used:

• Reciprocal identities, which relate a trigonometric function to its reciprocal, such as \(rac{1}{\sin \theta} = \csc \theta\).
• Quotient identities, which express one trigonometric function as a quotient of two others, like \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
• Pythagorean identities, most importantly \((\sin^2 \theta + \cos^2 \theta = 1)\), are draws from the Pythagorean Theorem.

In the solved exercise, the additive identity \(\sin(a + b) = \sin a \cos b + \cos a \sin b\) was crucial. This identity helps us find the sine of the sum of two angles, which was directly applied to the given expression after determining the individual sines and cosines using right triangles.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry, especially for right triangles. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed with the formula \(c^2 = a^2 + b^2\).
In practice, this theorem becomes valuable when you know the lengths of two sides and need to find the third. In our example, when given a triangle with sides 4 and 5, the Pythagorean Theorem quickly reveals the third side as 3 due to the equation \(5^2 = 4^2 + x^2\).
Recognizing such relationships can help you solve complex trigonometric expressions, as demonstrated in the exercise where we calculated the missing side lengths to determine the exact values of the sine and cosine for further simplification.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. These functions are defined as:

• Sine (\( \sin \theta \)) is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)) is the ratio of the length of the adjacent side to the hypotenuse.

These ratios are helpful in calculating unknown angles or sides in various scenarios involving right triangles. For the provided exercise, understanding and computing these ratios allowed us to use the additive formula effectively.
The ability to find the sine or cosine of a derived angle, like \( \cos^{-1} 4/5 \) or \( \sin^{-1} 3/5 \), showcases the theorem's potency in handling complex trigonometric evaluations. Knowing the relationships and how to manipulate sine and cosine values are key to completing many trigonometric operations, such as additive identity formulations.