Problem 61 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$(\sin x+\cos x)^{2}=1+\sin 2 x$$

Step-by-Step Solution

Answer

Identity is proven: Both sides simplify to \(1 + \sin 2x\).

Step 1: Expand the Left Side

First, we need to expand the left side of the equation \((\sin x + \cos x)^2\).Use the formula for a square of a sum:\((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = \sin x\) and \(b = \cos x\).So, \((\sin x + \cos x)^2 = \sin^2 x + 2 \sin x \cos x + \cos^2 x\).

Step 2: Simplify Using Pythagorean Identity

Notice that \(\sin^2 x + \cos^2 x = 1\) (Pythagorean Identity). Thus, we can simplify the expression: \(\sin^2 x + \cos^2 x + 2 \sin x \cos x = 1 + 2 \sin x \cos x\).

Step 3: Use the Double Angle Identity

The double angle identity for sine states that \( \sin 2x = 2 \sin x \cos x \).Therefore, we can replace \(2 \sin x \cos x\) with \(\sin 2x\) in our expression.So, \(1 + 2 \sin x \cos x = 1 + \sin 2x\).

Step 4: Conclude the Identity

Since both sides of the original equation simplify to the same expression \((1 + \sin 2x)\), the identity \((\sin x + \cos x)^2 = 1 + \sin 2x\) is proven.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the cornerstones in trigonometry. It is derived from the Pythagorean theorem, which relates the sides of a right triangle. This identity is written as \[\sin^2 x + \cos^2 x = 1\]This identity basically tells us that for any angle \(x\), the square of the sine of \(x\) plus the square of the cosine of \(x\) will always be equal to one.

The Pythagorean identity is incredibly useful in simplifying expressions during proof of identities, as seen in the exercise. In Step 2, the identity \(\sin^2 x + \cos^2 x = 1\) simplifies the expression obtained by expanding the left side of the equation.

Here’s how you can leverage this identity:

• Simplify complex trigonometric expressions.
• Convert between sine and cosine functions.
• Understand relationships within the unit circle since it represents the coordinates of any point on a circle with radius one.

Whenever you get stuck or see \(\sin^2 x + \cos^2 x\) in an equation, remember the Pythagorean identity—it might just be the key to simplifying your work!

double angle identity

The double angle identity provides a way to express trigonometric functions of double angles in terms of single angle functions:\[\sin 2x = 2 \sin x \cos x\]This identity is particularly useful when you need to simplify trigonometric expressions involving terms like \(2 \sin x \cos x\), as in Step 3 of the exercise.

In the problem, by recognizing the double angle identity, you can directly replace \(2 \sin x \cos x\) with \(\sin 2x\), thereby simplifying the equation significantly.

Mastering double angle identities opens up the possibility to:

• Transform products of sine and cosine into a more manageable form.
• Develop further trigonometric identities.
• Solve equations involving trigonometric functions effectively.

By converting the double angle into its equivalent single angle product, problems become simpler and straightforward. This insightful technique is essential for students aiming to cut through complex trigonometry challenges.

proof of identities

Proof of identities involves showing that two trigonometric expressions are equal for all values of the variable involved. The exercise exemplifies this process through a clear, step-by-step demonstration.

Here's a concise plan of action for proof of trigonometric identities:

• Start with one side: Choose one side of the equation, generally the more complex one, and aim to transform it to match the simpler side.
• Use known identities: Utilize known trigonometric identities like the Pythagorean or double angle identities to simplify expressions.
• Simplify adequately: Break down complex terms into manageable units.
• Conclude: After successfully transforming one side to the other, draw a conclusion that the identity holds.

In the example given, both the left side and the right side simplify to \(1 + \sin 2x\). Converging both sides to the same expression effectively proves the identity.

The skill of proving identities sharpens your understanding of trigonometric properties and enhances critical problem-solving abilities. It's like a puzzle—fitting all pieces together to demonstrate truth.