Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: Verify the identity. $$ \sin \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin \theta+\cos \theta) $$

Step-by-Step Solution

Answer

The identity is verified as both sides are equal.

Step 1: Apply the Angle Addition Formula

The identity involves the sine of a sum, \(\theta + \frac{\pi}{4}\). Use the angle addition formula for sine: \(\sin(a + b) = \sin a \cos b + \cos a \sin b.\) Let \(a = \theta\) and \(b = \frac{\pi}{4}\).

Step 2: Calculate Sine and Cosine of \(\frac{\pi}{4}\)

Values for trigonometric functions at \(\frac{\pi}{4}\) are known: \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}.\)

Step 3: Substitute into the Angle Addition Formula

Substitute \(\sin \frac{\pi}{4}\) and \(\cos \frac{\pi}{4}\) into the formula: \[\sin \left(\theta + \frac{\pi}{4}\right) = \sin \theta \cdot \cos \frac{\pi}{4} + \cos \theta \cdot \sin \frac{\pi}{4}. \]This becomes: \[\sin \left(\theta + \frac{\pi}{4}\right) = \sin \theta \cdot \frac{\sqrt{2}}{2} + \cos \theta \cdot \frac{\sqrt{2}}{2}.\]

Step 4: Factor Out \(\frac{\sqrt{2}}{2}\)

Notice that both terms on the right side of the equation have \(\frac{\sqrt{2}}{2}\) as a factor. Factor it out: \[\sin \left(\theta + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\sin \theta + \cos \theta).\]

Step 5: Conclusion

We have shown that both sides of the given identity are equal: \[\sin \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin \theta+\cos \theta).\]Thus, the identity is verified.

Key Concepts

Angle Addition Formula

The angle addition formula is a powerful tool in trigonometry that helps us evaluate the sine, cosine, and tangent of the sum or difference of two angles. Here, we focus on the angle addition formula for the sine function. It states:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

This formula is derived from the properties of the unit circle and is essential when tackling problems involving compound angles. In this context, if we know the values of \( a \) and \( b \), and individual trigonometric function values, we can find the sine of \( a + b \) by applying this formula.
For example, when verifying an identity like \( \sin(\theta + \frac{\pi}{4}) \), the angle addition formula allows us to express it in terms of \( \sin \theta \) and \( \cos \theta \), using known values for \( \sin \left(\frac{\pi}{4}\right) \) and \( \cos \left(\frac{\pi}{4}\right) \). This formula is not just limited to sine, but there are similar versions for cosine and tangent functions too.

Sine Function

The sine function is one of the most fundamental trigonometric functions and is typically abbreviated as \( \sin \). It relates an angle in a right triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. This function is periodic with a period of \( 2\pi \), reflecting its repetition every full circle around the unit circle.

• The sine of a 45-degree angle (or \( \frac{\pi}{4} \) radians) is \( \frac{\sqrt{2}}{2} \).
• This is a critical value often used in trigonometric calculations.

The sine function is continuous and smooth, and part of its charm lies in how it changes value as the angle increases, reflecting a wave-like pattern when graphed over its domain. When dealing with sum identities, such as those involving angle additions, this function's properties allow for transformation based on known angles, as illustrated in the given identity verification.
Remember, the complement of sine — found using angle difference forms or sine equalities — carries through seamlessly due to the symmetry in the unit circle.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The primary trig functions are sine, cosine, and tangent, often introduced in school and used extensively in physics, engineering, and more advanced mathematical fields.

• They are defined based on the unit circle, where the radius is 1.
• Each function has specific identities and properties, such as periodicity and symmetry.

In particular, cosine and sine play crucial roles in angle addition and subtraction identities and are especially useful in solving problems involving wave patterns, circular motion, and oscillations.
They enable us to simplify expressions and verify identities, such as \( \sin(\theta + \frac{\pi}{4}) \), by using properties like even and odd functionalities. Trigonometric functions hold a vast repertoire of identities, helping bridge purely algebraic expressions with geometric intuitions.