Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\sin ^{2} 26^{\circ}+\cos ^{2} 26^{\circ}$$

Short answer

1

Step-by-step solution

Identify the fundamental trigonometric identity

Recall the Pythagorean identity, which states that for any angle \(\theta\), \(\sin^2(\theta) + \cos^2(\theta) = 1\).

Apply the identity to the given angle

Here, the given angle is 26°. Substitute \(26^{\circ}\) for \(\theta\) in the identity: \(\sin^2(26^{\circ}) + \cos^2(26^{\circ})\).

Simplify the expression

Using the Pythagorean identity, we find that \(\sin^2(26^{\circ}) + \cos^2(26^{\circ}) = 1\).

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental equations in trigonometry that relate the angles and sides of triangles. These identities are vital tools in simplifying trigonometric expressions and solving trigonometric equations. Some of the most common trigonometric identities include:

• Pythagorean identities
• Reciprocal identities
• Quotient identities

The Pythagorean identity is particularly important and states that for any angle \( \theta \), the equation \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] holds true. This identity is derived from the Pythagorean theorem in the context of a right-angled triangle. Another key identity is the complementary angle identity, which relates the sine of an angle to the cosine of its complement.

Complementary Angle Theorem

The complementary angle theorem is a useful concept in trigonometry. It states that the sine of an angle is equal to the cosine of its complement. Remember that two angles are complementary if their sum is 90 degrees. Mathematically, for any angle \( \theta \), the theorem can be written as:

• \[ \sin(\theta) = \cos(90^\circ - \theta) \]
• \[ \cos(\theta) = \sin(90^\circ - \theta) \]

For example, if \( \theta = 30^\circ \), the complementary angle is \( 60^\circ \). This means:\[ \sin(30^\circ) = \cos(60^\circ) \]\[ \cos(30^\circ) = \sin(60^\circ) \]The complementary angle theorem can simplify calculations and solve trigonometric equations involving complementary angles.

Exact Trigonometric Values

Exact trigonometric values are the precise, often non-decimal values of trigonometric functions at specific angles. Some common angles for which we often know these exact values include 0°, 30°, 45°, 60°, and 90°. For instance:

• \[ \sin(0^\circ) = 0 \]
• \[ \cos(0^\circ) = 1 \]
• \[ \sin(30^\circ) = \frac{1}{2} \]
• \[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
• \[ \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \]

Knowing these exact values helps in solving trigonometric equations and verifying identities without the need for a calculator. In the given exercise, we use such exact values and identities to simplify the expression without any numerical approximation, ensuring precision in our solutions.