Problem 33

Question

Verify that each of the following is an identity. $\cos (2 \pi+\theta)=\cos \theta$

Step-by-Step Solution

Verified

Answer

The identity $ \cos(2\pi + \theta) = \cos \theta $ is verified as true.

1Step 1: Understand the Identity Format

We need to verify whether the given expression $ \cos(2\pi + \theta) = \cos \theta $ is true for all values of $ \theta $. This means proving that the left side equals the right side.

2Step 2: Apply the Angle Sum Formula for Cosine

Recall the angle sum formula for cosine: $ \cos(a+b) = \cos a \cdot \cos b - \sin a \cdot \sin b $. Here, set $ a = 2\pi $ and $ b = \theta $.

3Step 3: Substitute Values for Cosine and Sine of 2π

We know that $ \cos 2\pi = 1 $ and $ \sin 2\pi = 0 $. Substitute these into the formula: \[ \cos(2\pi + \theta) = \cos 2\pi \cdot \cos \theta - \sin 2\pi \cdot \sin \theta \] This simplifies to: \[ 1 \cdot \cos \theta - 0 \cdot \sin \theta = \cos \theta \]

4Step 4: Simplify the Expression

As shown in the previous step, the expression simplifies to just $ \cos \theta $, confirming that both sides of the original equation are indeed equal.

5Step 5: Conclusion: Verification

The given identity $ \cos(2\pi + \theta) = \cos \theta $ holds true as we have shown both sides equal after simplification with trigonometric identities.

Key Concepts

Angle Sum FormulaCosine FunctionUnit CircleSine Function

Angle Sum Formula

The Angle Sum Formula is a foundational tool in trigonometry that helps us simplify the cosine and sine values when an angle is expressed as the sum of two other angles. You can think of it as a recipe for breaking down complicated angles into smaller, more manageable parts.
For the cosine of the sum of two angles, the formula is:

$\cos(a+b) = \cos a \cdot \cos b - \sin a \cdot \sin b$

This formula is particularly useful when working with angles like $2\pi$ plus another angle, $\theta$. By using this formula, we can express the cosine of a sum as a combination of the cosines and sines of the original angles. This approach allows us to simplify complex trigonometric expressions into simpler, equivalent forms.
So whenever you're faced with an angle sum, remember this formula - it's your key to unraveling trigonometric puzzles!

Cosine Function

The cosine function, represented as $ \cos $, is one of the primary functions in trigonometry. It quantifies the horizontal position of a point on the unit circle corresponding to a given angle.
This means when an angle $\theta$ is plotted on the unit circle, $\cos \theta$ provides the x-coordinate of the point on the circle.

$\cos(0) = 1$ because the point at an angle of $0$ degrees (or $0$ radians) is on the rightmost side of the unit circle.
Similarly, $\cos(2\pi) = 1$ since $2\pi$ brings us full circle back to the point where we started.

Knowing the cosine function's values at different key angles is crucial for solving trigonometric problems. The function repeats every $2\pi$, so if you shift an angle by a full circle ($2\pi$), $\cos \theta$ remains the same. This property makes the cosine function periodic and predictable, which is frequently used in trigonometry to verify identities and solve equations.

Unit Circle

The unit circle is a simple yet powerful conceptual tool in trigonometry. It's a circle centered at the origin of a coordinate plane with a radius of one unit. The power of the unit circle lies in its ability to easily demonstrate the relationships between trigonometric functions and the coordinates of points on the circle.
The unit circle helps visualize how cosine and sine values are derived:

Each angle $\theta$ on the circle corresponds to a specific point $(\cos \theta, \sin \theta)$.
This makes it a perfect reference for understanding the periodic nature of trig functions, particularly since each full revolution ($2\pi$ radians) around the circle brings you back to the starting point.

In the context of the cosine identity $ \cos(2\pi + \theta) = \cos \theta $, understanding the unit circle's role is vital. Since going around the circle adds $2\pi$ to your angle without changing the coordinates $(\cos \theta, \sin \theta)$, the values of the cosine and sine functions are preserved, illustrating how seamlessly it all fits together in trigonometric identities.

Sine Function

The sine function, $ \sin $, often pairs with the cosine function as a trigonometric counterpart. It measures the vertical component of an angle positioned on the unit circle. When you plot $\theta$ on the unit circle, $\sin \theta$ gives the y-coordinate of that point.
The sine of commonly known angles demonstrates its periodic nature:

$\sin(0) = 0$ because the point is at the origin height-wise.
$\sin(2\pi) = 0$ also takes place because $2\pi$ is a full circle completed, effectively bringing the point back to the starting horizontal line.

This periodic cycle means that, like cosine, the sine function repeats every $2\pi$ radians. Understanding that $\sin 2\pi = 0$ aids in breaking down angle sum problems, as seen in the identity $ \cos(2\pi + \theta) = \cos \theta $, where $\sin 2\pi$ multiplies the sine of the angle $\theta$, effectively simplifying the equation by eliminating the term. Knowing these core functions and how they interrelate is key to mastering trigonometry.

Problem 33

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