Problem 65 - Precalculus with Limits

Question

In Exercises 65-68, use the co-function identities to evaluate the expression without using a calculator. \( \sin^2 25^\circ + \sin^2 65^\circ \)

Step-by-Step Solution

Answer

The solution of the expression \( \sin^2 25^\circ + \sin^2 65^\circ \) is 1.

Step 1: Recognize identity

Firstly, the above exercise can be written in terms of sine and cosine according to co-function identity \(\sin(90° - a) = \cos(a)\). So, \(\sin^2 65°\) can be rewritten as \(\cos^2 25°\). Hence the expression becomes \(\sin^2 25° + \cos^2 25°\).

Step 2: Use identity

In the second step, apply the Pythagorean identity concerning sine and cosine which states that \(\sin^2(a) + \cos^2(a) = 1\). This identity is true for any angle a, thus in this case a = 25°. So the expression \(\sin^2 25° + \cos^2 25°\) simplifies to 1.

Key Concepts

Co-function Identity

The co-function identities are essential tools when you work with trigonometric functions. In simple terms, they relate the sine and cosine functions at complementary angles. The word 'co-function' implies a relationship where these functions complement each other. Just remember that complementary angles are two angles that add up to 90 degrees.

For example, the identity \( \sin(90° - a) = \cos(a) \) shows us how the sine of an angle can be related to the cosine of its complement. When solving the original exercise, we used this identity to replace \( \sin^2 65° \) with \( \cos^2 25° \). This is because \( 65° \) and \( 25° \) are complementary angles, i.e., they sum up to 90°. Using co-function identities helps transform trigonometric expressions, making them easier to simplify or evaluate without a calculator.

• Co-function identities transform sine and cosine with complementary angles.
• Useful for simplification and problem-solving without a calculator.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental relationships in trigonometry. It establishes that for any angle \( a \), \( \sin^2(a) + \cos^2(a) = 1 \). This identity comes from the Pythagorean theorem, applying it to the unit circle.

In our exercise, after using the co-function identity, we were left with the expression \( \sin^2 25° + \cos^2 25° \). By applying the Pythagorean identity, this sum simplifies directly to 1, thanks to its universal truth for any angle.

• Based on the Pythagorean theorem in the unit circle.
• \( \sin^2(a) + \cos^2(a) = 1 \) holds true for all angles \( a \).
• This identity frequently simplifies trigonometric expressions.

Sine and Cosine Functions

Sine and cosine are foundational trigonometric functions that describe relationships in a right triangle and beyond. These functions are not just limited to right triangles but extend to describe wave-like patterns in the unit circle.

The sine of an angle corresponds to the y-coordinate of a point on the unit circle, while the cosine represents the x-coordinate. This relationship is at the heart of trigonometry, appearing in co-function and Pythagorean identities as shown in the exercise above.

• Sine and cosine are key in describing circular and periodic phenomena.
• They provide the base for understanding more complex trigonometric identities.
• Using these functions effectively can simplify seemingly complex problems.