Problem 34 - Precalculus with Calculus Previews

Question

Justify the given statement with one of the properties of the trigonometric functions. $$ \cos (\pi / 4)=\sin (\pi / 4) $$

Step-by-Step Solution

Answer

The equality holds due to trigonometric values and the complementary angle identity.

Step 1: Recall Trigonometric Values

The standard values for trigonometric functions at common angles are essential. Specifically, \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Recognizing these values will help us to show that the given statement is true.

Step 2: Using Complementary Angle Identity

The equality \( \cos(\theta) = \sin(\pi/2 - \theta) \) can be used. Substitute \( \theta = \pi/4 \): \( \cos(\pi/4) = \sin(\pi/2 - \pi/4) = \sin(\pi/4) \). Thus, \( \cos(\pi/4) = \sin(\pi/4) \) by the complementary angle identity.

Step 3: Verify Equality

Verify that both \( \cos(\pi/4) \) and \( \sin(\pi/4) \) give the same result, \( \frac{\sqrt{2}}{2} \). Hence, the statement \( \cos(\pi/4) = \sin(\pi/4) \) holds true.

Key Concepts

Complementary Angle Identity

One fascinating aspect of trigonometry lies in the relationships between different trigonometric functions, especially through identities like the complementary angle identity. The complementary angle identity states that \(\cos(\theta) = \sin(\frac{\pi}{2} - \theta)\). This means that the cosine of any angle is equal to the sine of its complement, where the complement is \(90\) degrees or \(\pi/2\) radians minus the original angle.

In the given exercise, to prove \(\cos(\pi/4) = \sin(\pi/4)\), we utilized this identity. We must remember that \(\pi/4\) is equivalent to \(45\) degrees, and its complementary angle within the first quadrant is also \(45\) degrees, because \(90^{\circ} - 45^{\circ} = 45^{\circ}\). Therefore, \(\cos(\pi/4)\) is equal to \(\sin(\pi/4)\).

This reliable identity simplifies evaluating and proving equalities between sine and cosine functions, especially when dealing with angles expressed in radians. It’s a useful tool in both mathematics and physics, where angle transformations often occur.

Standard Trigonometric Values

Standard trigonometric values are fundamental when working with angles in trigonometry, particularly at common angles such as \(0, \pi/6, \pi/4, \pi/3, \) and \(\pi/2\). Recognizing these can significantly speed up exercises and provide insight when evaluating trigonometric functions:

• \(\sin(\pi/6) = \frac{1}{2}\)
• \(\sin(\pi/4) = \frac{\sqrt{2}}{2}\)
• \(\cos(\pi/4) = \frac{\sqrt{2}}{2}\)
• \(\sin(\pi/3) = \frac{\sqrt{3}}{2}\)
• \(\cos(\pi/3) = \frac{1}{2}\)

In this exercise, we specifically used \(\cos(\pi/4)\) and \(\sin(\pi/4)\), both equivalent to \(\frac{\sqrt{2}}{2}\). The equal outcome of these evaluations proves the statement \(\cos(\pi/4) = \sin(\pi/4)\).

Memorizing these values, especially for angles like \(\pi/4\), is invaluable as it aids not only in problem-solving but also deepens understanding of trigonometric functions and their interrelationships.

Trigonometric Equalities

Trigonometric equalities allow us to relate different trigonometric functions to each other, thereby verifying mathematical statements about angles and their functions. They are foundational to proving trigonometric identities and solving trigonometric equations. In the given problem, the equality \(\cos(\pi/4) = \sin(\pi/4)\) not only draws upon the standard trigonometric values but also highlights the concept of inter-function equality through identities.

Using identities such as \(\cos(\theta) = \sin(\pi/2 - \theta)\), we can show that two different functions yield identical results based on their complementary relationship. This is true especially when evaluating functions at symmetrical angles around \(\pi/4\) and \(\pi/2\).

Understanding and applying these equalities is vital in various fields such as engineering, physics, and computer graphics, where such functions frequently model waves, oscillations, and rotational forces.