Problem 25 - Algebra 2

Question

Verify that each of the following is an identity. $$ \sin ^{2} x=\frac{1}{2}(1-\cos 2 x) $$

Step-by-Step Solution

Answer

The identity \( \sin^{2} x = \frac{1}{2}(1 - \cos 2x) \) is verified as true for all \( x \).

Step 1: Understand the Identity

The identity given is \( \sin^{2} x = \frac{1}{2}(1 - \cos 2x) \). We need to show that both sides of the equation are equal for all values of \( x \). Essentially, verify that the left side (\( \sin^{2} x \)) can be transformed into the right side or vice versa.

Step 2: Use Double Angle Formula for Cosine

Recall the double angle formula for cosine: \( \cos 2x = 1 - 2\sin^{2} x \). This formula will help us relate \( \sin^{2} x \) to \( \cos 2x \).

Step 3: Substitute into the Identity

Using the double angle formula, substitute \( \cos 2x = 1 - 2\sin^{2} x \) into the right side of the identity: \[ \frac{1}{2}(1 - \cos 2x) = \frac{1}{2}(1 - (1 - 2\sin^{2} x)) \].

Step 4: Simplify the Expression

Simplify the expression inside the parentheses:\[ 1 - (1 - 2\sin^{2} x) = 1 - 1 + 2\sin^{2} x = 2\sin^{2} x \].

Step 5: Final Simplification

Now, divide the expression by 2:\[ \frac{1}{2} \times 2\sin^{2} x = \sin^{2} x \]. This shows the left side, \( \sin^{2} x \), is equal to the right side, \( \frac{1}{2}(1 - \cos 2x) \).

Step 6: Conclusion

We have demonstrated that \( \sin^{2} x = \frac{1}{2}(1 - \cos 2x) \) by simplifying both sides to be equal, confirming that the identity holds true.

Key Concepts

Double Angle Formulas

Double angle formulas are powerful tools in trigonometry that help simplify expressions and solve equations involving angles. These formulas relate the trigonometric functions of an angle to its double. For example, the double angle formula for cosine is:\[\cos 2x = 1 - 2\sin^2 x\]This formula links the cosine of double an angle (\cos 2x) to the sine of the original angle (\sin x). These identities are particularly useful in proofs and simplifying trigonometric expressions.

In our exercise, we used the double angle formula for cosine to transform and verify the given identity:\[\sin^2 x = \frac{1}{2}(1 - \cos 2x)\]Recognizing and applying double angle formulas can greatly simplify otherwise complex trigonometric equations, and are an essential part of trigonometric analysis. They are used not only to verify identities but also in calculus and solving problems involving periodic functions.

Trigonometric Functions

Trigonometric functions are fundamental mathematical functions that describe the relationships between the angles and sides of right-angled triangles. The main functions are sine (\sin), cosine (\cos), and tangent (\tan). Each of these functions has its specific geometric meaning and can also be represented on the unit circle.

• \(\sin x\) is the vertical coordinate of a point on the unit circle.
• \(\cos x\) is the horizontal coordinate.
• \(\tan x\) is the ratio of sine to cosine.

In verifying trigonometric identities, like the one in our exercise, understanding these functions is crucial. You need to visualize how these functions interact and how their equations can represent specific angles' relationships.

Moreover, these functions extend beyond triangles and are essential in modeling periodic phenomena such as sound waves, alternating current, and tides in physics.

Trigonometric Simplification

Trigonometric simplification involves reducing trigonometric expressions to simpler forms. This is often necessary to verify identities or solve trigonometric equations. By using known identities, like the Pythagorean identity or double angle formulas, we can transform and simplify these functions.In our example, we transformed the complex trigonometric expression using the double angle cosine formula to verify the identity:\[\sin ^2 x = \frac{1}{2}(1 - \cos 2x)\]Simplification typically involves a few key steps:

• Use identities to replace complex parts of the equation.
• Expand and simplify expressions, such as distributing multipliers.
• Combine like terms and reduce fractions.

These techniques not only help in verifying identities but also make solving trigonometric equations more manageable. Becoming proficient at simplification enables you to tackle more advanced problems with confidence.