Problem 65 - Algebra and Trigonometry

Question

Find the exact value of each expression. Do not use a calculator. $$ 1+\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ} $$

Step-by-Step Solution

Answer

The exact value of the expression \(1+\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}\) is 2.

Step 1: Converting sin to cos

Firstly, recognize that 50 degrees is the complement of 40 degrees, therefore you can convert \(\sin^{2}(50^{\circ})\) to \(\cos^{2}(40^{\circ})\). This leads us to: \(1 + \sin^{2}(40^{\circ}) + \cos^{2}(40^{\circ})\)

Step 2: Use the Pythagorean trigonometric identity

The transformed expression in the first step, \(1 + \sin^{2}(40^{\circ}) + \cos^{2}(40^{\circ})\), can further be simplified using the Pythagorean trigonometric identity. This gives: \(1 + 1 = 2\)

Key Concepts

Complementary Angles

Complementary angles are pairs of angles that add up to 90 degrees. This is a special relationship in trigonometry that allows us to swap sine and cosine functions because they are, in essence, reflections in the unit circle. For example, if you have an angle of 40 degrees, its complementary angle is 50 degrees because 40 + 50 = 90.

This complementary relationship is crucial when working with trigonometric identities, as you can swap the sine of one angle for the cosine of its complementary angle and vice versa. Therefore, \(\sin(50^{\circ}) = \cos(40^{\circ})\). This is especially helpful for simplifying trigonometric expressions where directly calculating values might be cumbersome otherwise.

In the context of the original exercise, recognizing that 50 degrees and 40 degrees are complementary allowed a significant simplification by substituting sine with cosine. This recognized substitution is a key step in leveraging complementary angles for simplification.

Sine and Cosine Relationship

The sine and cosine functions are closely linked through the concept of complementary angles as well as through the Pythagorean identity. In a right triangle or the unit circle, these functions represent the opposite and adjacent sides over the hypotenuse, respectively.

A core relationship between sine and cosine can be observed through the identity: for any angle \( \theta \), the sine of \(\theta\) is equal to the cosine of its complement, which is \(90^{\circ} - \theta\). This means \(\sin(\theta) = \cos(90^{\circ} - \theta)\) and \(\cos(\theta) = \sin(90^{\circ} - \theta)\).

This relationship was used in the exercise to convert \(\sin^2(50^{\circ})\) into \(\cos^2(40^{\circ})\). Understanding these relationships not only helps in simplifying expressions but also forms the basis for solving many trigonometric equations and problems efficiently.

Trigonometric Simplification

Trigonometric simplification often involves using trigonometric identities to make expressions more manageable. In the original exercise, simplifying the expression was achieved through two main techniques: recognizing complementary angles and applying the Pythagorean identity.

Once we converted \(\sin^2(50^{\circ})\) to \(\cos^2(40^{\circ})\), the next step was to utilize the Pythagorean identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\). This means for any angle \( \theta \), the sum of the squares of sine and cosine of that angle is always 1. Thus, in the expression \( 1 + \sin^2(40^{\circ}) + \cos^2(40^{\circ}) \), recognizing the identity simplifies it directly to 2.

By understanding trigonometric simplification, you can transform complex expressions into simpler forms, making problem-solving more straightforward while ensuring computations remain accurate and efficient.