Problem 15 - Precalculus: Functions and Graphs

Question

Verify the identity. \((\sin t+\cos t)^{2}=1+\sin 2 t\)

Step-by-Step Solution

Answer

The identity is verified as both sides equal 1 + sin 2t.

Step 1: Expand the Left Side

Start by expanding \((\sin t + \cos t)^{2}\) using the distributive property: \[ (\sin t + \cos t)^{2} = (\sin t + \cos t)(\sin t + \cos t) \]Apply the distributive property (FOIL method):\[ \sin^2 t + 2\sin t\cos t + \cos^2 t \]

Step 2: Use Pythagorean Identity

Use the Pythagorean identity for sine and cosine:\[ \sin^2 t + \cos^2 t = 1 \]Substitute this identity into the expression to simplify:\[ 1 + 2\sin t\cos t \]

Step 3: Use Double Angle Identity

Use the double angle identity for sine:\[ \sin 2t = 2\sin t\cos t \]Replace \(2\sin t\cos t\) with \(\sin 2t\) in the expression:\[ 1 + \sin 2t \]

Step 4: Conclusion

Since both sides of the equation are now equal, i.e.,\[ (\sin t + \cos t)^{2} = 1 + \sin 2t \]we have verified the identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine functions. It states that for any angle \( t \), the relationship \( \sin^2 t + \cos^2 t = 1 \) always holds. This identity is rooted in the Pythagorean Theorem, which deals with the relationship between the sides of a right-angled triangle. In the context of the unit circle, where the radius is 1, this theorem shows how the sine and cosine of an angle correspond to the coordinates of a point on the circle. By understanding this identity, we recognize that no matter the angle, the sum of the squares of the sine and cosine will always equal 1. This concept is crucial as it allows us to simplify expressions and verify various trigonometric identities, like the one provided in the exercise.

Double Angle Identity

The double angle identity is a useful formula in trigonometry, which relates trigonometric functions of double angles to functions of single angles. Specifically, for the sine function, the double angle identity is given by \( \sin 2t = 2 \sin t \cos t \). This identity helps in simplifying expressions where an angle is doubled, making it easier to work with trigonometric equations.

• Simplification: It allows us to replace a term like \( 2 \sin t \cos t \) with \( \sin 2t \), making complex expressions simpler.
• Problem-solving: It's particularly useful in solving equations and verifying identities where double angles are involved.

Understanding this identity equips students to tackle a variety of trigonometric problems with greater ease, including the verification of identities like the one in this exercise.

Distributive Property

The distributive property is a basic algebraic principle that plays a significant role in expanding expressions, particularly polynomials. It states that for any numbers or expressions \( a, b, \) and \( c \), the equation \( a(b + c) = ab + ac \) holds. In trigonometry, this property is often used for expanding expressions involving trigonometric terms. For instance, in the given exercise, we applied the distributive property, sometimes called the FOIL method when expanding binomials, to expand \( (\sin t + \cos t)^2 \). This involves:

• Multiplying \( \sin t \times \sin t \)
• Adding \( \sin t \times \cos t \) and \( \cos t \times \sin t \)
• Finally adding \( \cos t \times \cos t \)

The result is \( \sin^2 t + 2\sin t\cos t + \cos^2 t \). This expansion is the starting point for verifying the identity given in the exercise. Understanding and applying the distributive property is essential for manipulating and simplifying trigonometric expressions.