Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \sin 10 \theta=2 \sin 5 \theta \cos 5 \theta $$

Step-by-Step Solution

Answer

The identity is verified using the double angle formula.

Step 1: Recall the double angle formula

The double angle formula for sine states: \(\sin 2\alpha = 2\sin\alpha\cos\alpha\).

Step 2: Apply with α = 5θ

Let \(\alpha = 5\theta\). Then \(\sin(2 \cdot 5\theta) = 2\sin 5\theta \cos 5\theta\), i.e., \(\sin 10\theta = 2\sin 5\theta \cos 5\theta\). This verifies the identity.

Key Concepts

Sine Function

The sine function is one of the primary functions in trigonometry. It relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. For any angle \( \theta \), the sine function can be represented using: \[ \sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}} \] Sine function values repeat every \( 360^{\circ} \) or \( 2\pi \) radians, which means the sine function is periodic.

• The sine function is an odd function, meaning that \( \sin(-\theta) = -\sin(\theta) \).
• At \( \theta = 0 \), the sine value is 0.
• It reaches its maximum value of 1 at \( \theta = 90^{\circ} \) or \( \pi/2 \) radians.
• The minimum value is -1, which occurs at \( \theta = 270^{\circ} \) or \( 3\pi/2 \) radians.

Understanding these properties is essential, especially when verifying trigonometric identities like the one in the exercise, as we'll use the behavior of the sine function frequently in calculations.

Cosine Function

The cosine function is another fundamental trigonometric function, complementing the sine function. It pairs an angle with the ratio of the adjacent side over the hypotenuse in a right triangle: \[ \cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \] Just like sine, the cosine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians.

• The cosine function is an even function, which means \( \cos(-\theta) = \cos(\theta) \).
• For \( \theta = 0 \), the cosine value is 1.
• The cosine function value is 0 at \( \theta = 90^{\circ} \) or \( \pi/2 \) radians.
• The minimum value of -1 occurs at \( \theta = 180^{\circ} \) or \( \pi \) radians.

The cosine function plays a crucial role in our identity verification, particularly as part of the product \( 2 \sin 5\theta \cos 5\theta \). Recognizing its symmetry and periodicity helps us simplify and verify complex trigonometric identities.

Verification of Identities

Verification of identities is an essential skill in trigonometry. It involves proving that two expressions are equivalent for all values of the occurring variables (angles, in this case). In our exercise, we are verifying that \( \sin 10\theta = 2 \sin 5\theta \cos 5\theta \).

• Start by choosing to manipulate either the expression on the LHS or RHS. Here, we often prefer to manipulate the more complex side to match the simpler one.
• Use known identities or formulas, like the angle addition or double angle formulas, to simplify the expression.
• Remember that each step must logically follow from the last; showing continuity in logical reasoning.

The ultimate goal is to transform one side into the other completely. Practicing the verification of identities enhances understanding of trigonometric properties and offers alternative ways to approach problems.

Angle Multiplication Formulas

Angle multiplication formulas are special trigonometric identities that allow us to express trigonometric functions of multiplied angles, like \( n\theta \), in terms of functions of \( \theta \). These formulas include double angle formulas such as:

• For sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• For cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

Similarly, when you have multiples of an odd integer like in our exercise, you often use product-to-sum identities. For example: \[ \sin m\theta = 2 \sin \frac{m}{2} \theta \cos \frac{m}{2} \theta \text{ (when \( m \) is even)} \] These identities are crucial to manipulate and simplify expressions, allowing us to prove or disprove equivalencies in complex trigonometric equations. Understanding and applying these can make the process of identity verification much more straightforward.