Question

Show \(\cos \left(360^{\circ}-\theta\right)=\cos \theta\)

Short answer

Question: Prove that the cosine of an angle difference (360° - θ) is equal to the cosine of the original angle (θ). Answer: Based on our calculations, we proved that \(\cos(360^{\circ} - \theta)=\cos(\theta)\) using the angle difference formula for cosine.

Step-by-step solution

Write the angle difference formula for cosine

Recall that the angle difference formula for cosine is: \(\cos(a - b)=\cos a \cos b + \sin a \sin b\).

Substitute the given values into the formula

Let's substitute \(a=360^{\circ}\) and \(b=\theta\) into the formula: \(\cos(360^{\circ} - \theta)=\cos(360^{\circ})\cos(\theta) + \sin(360^{\circ})\sin(\theta)\).

Evaluate the cosine and sine of 360°

We know that \(\cos(360^{\circ})=1\) and \(\sin(360^{\circ})=0\). Let's substitute these values into the expression: \(\cos(360^{\circ} - \theta)=1\cdot\cos(\theta) + 0\cdot\sin(\theta)\).

Simplify the expression

Now, we simplify the expression: \(\cos(360^{\circ} - \theta)=\cos(\theta)\). Therefore, we have demonstrated that \(\cos(360^{\circ} - \theta)=\cos(\theta)\).

Key concepts

Angle Difference Formula

The angle difference formula is a vital trigonometric identity that helps to find the cosine of the difference between two angles. It is expressed as: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] This formula is especially useful when you encounter angles that are not standard or do not directly appear on the unit circle. It allows one to express these angles in terms we know, like calculations involving angles such as \(360^{\circ}\) and \(\theta\). For the exercise given, the formula was applied to \(\cos(360^{\circ} - \theta)\), revealing a step-by-step approach to simplifying and finding its value.

Cosine Function

The cosine function, often denoted as \( \cos \theta \), arises in trigonometry as one of the primary trigonometric functions. It relates to the horizontal coordinate on the unit circle for a given angle \( \theta \). In many practical scenarios, the cosine function expresses periodic phenomena, such as waves.

• \( \cos(0^{\circ}) = 1 \)
• \( \cos(90^{\circ}) = 0 \)
• \( \cos(180^{\circ}) = -1 \)
• \( \cos(360^{\circ}) = 1 \)

In the example, knowing \( \cos(360^{\circ}) = 1 \) greatly simplifies the application of the angle difference formula.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Any point on this circle can be expressed using sine and cosine.

• X-coordinate: \( \cos \theta \)
• Y-coordinate: \( \sin \theta \)

Using the unit circle, you can easily determine the sine and cosine values of commonly used angles. For example, at \( 360^{\circ} \), the coordinates return to \((1, 0)\), denoting \( \cos(360^{\circ}) = 1 \) and \( \sin(360^{\circ}) = 0 \). Understanding the unit circle can also explain why angles like \( 360^{\circ} - \theta \) resort back to equivalent angles like \( \theta \).

Trigonometric Simplification

Trigonometric simplification involves reducing expressions to make them easier to understand or compute. It often involves applying identities like the angle difference formula, Pythagorean identities, and recognizing angle equivalences. In the specific example of \( \cos(360^{\circ} - \theta) = \cos \theta \), simplification starts with substituting known values: \( \cos(360^{\circ}) = 1 \) and \( \sin(360^{\circ}) = 0 \). The process results in: \[ \cos(360^{\circ} - \theta) = 1 \cdot \cos(\theta) + 0 \cdot \sin(\theta) = \cos(\theta) \]This reduces the problem and provides the final simplified result without further computation. Mastering trigonometric simplification can thus make solving extensive problems much more manageable.