Question

Explain how you would find the value of \(\cos \left(-45^{\circ}\right)\) using even-odd properties.

Short answer

\( \cos(-45^{\box{circ}}) = \cos(45^{\box{circ}}) = \frac{\sqrt{2}}{2} \)

Step-by-step solution

Understanding Trigonometric Even-Odd Properties

Recall that cosine is an even function. This means that \(\begin{align*} \cos(-x) = \cos(x) \).

Applying the Property to \( -45^{\box{circ}} \)

Using the even property of cosine, substitute \( x = -45^{\box{circ}} \). Therefore, \( \cos(-45^{\box{circ}}) = \cos(45^{\box{circ}}) \).

Evaluate \( \cos(45^{\box{circ}}) \)

Recall from basic trigonometric values that \(\begin{align*} \cos(45^{\box{circ}}) = \frac{\sqrt{2}}{2} \). Hence, \( \cos(-45^{\box{circ}}) = \frac{\sqrt{2}}{2} \).

Key concepts

even-odd properties

Trigonometric functions have specific symmetry properties, known as even-odd properties. These properties help simplify the evaluation of trigonometric functions at negative angles. A function is considered 'even' if \( f(-x) = f(x) \) for all values of \( x \). This means that the value of the function at a negative input is the same as the value at the positive input.

Cosine is one of these even functions. Knowing that cosine is even, you can confidently state that \( \begin{align*} \cos(-x) = \cos(x) \)\.

This property becomes extremely useful when you need to find the cosine of a negative angle, as it relieves you from the need to directly evaluate the function at that negative angle. Instead, you can evaluate it at the corresponding positive angle, simplifying the calculation immensely.

Remember: Sine, on the other hand, is an odd function, which means \( \sin(-x) = -\sin(x) \), making its behavior different from that of the cosine function.

cosine function

The cosine function is one of the fundamental trigonometric functions and is often denoted as \( \cos(x) \)\. It represents the adjacent side over the hypotenuse in a right-angled triangle.

Mathematically, for an angle \( x \) in a right triangle, \( \cos(x) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

In the unit circle, \( \cos(x) \) also represents the x-coordinate of a point lying on the circle, making it useful for many applications involving periodic phenomena like waves.

Because cosine is an even function, it has symmetry around the y-axis. This means that \( \cos(-45^{\circ}) = \cos(45^{\circ}) \) because the function value at a negative angle equals the function value at the positive angle.

For example, knowing this makes calculating trigonometric values easier, as you only need to remember the positive angles.

basic trigonometric values

Understanding basic trigonometric values is essential for evaluating trigonometric functions at common angles. One crucial value to remember is \( \cos(45^{\circ}) \).

The angle of 45 degrees, or \( \frac{\pi}{4} \) radians, is special in trigonometry because it appears frequently in various applications and problems.

To find \( \cos(45^{\circ}) \), recall that for an isosceles right triangle (45-45-90 triangle), the lengths of the legs are equal, and the hypotenuse is \( \sqrt{2} \) times longer than each leg.

Therefore, \( \cos(45^{\circ}) = \frac{1}{\sqrt{2}} \), which is often rationalized to \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \).

Knowing these fundamental values helps simplify solving trigonometric problems. For \( \cos(-45^{\circ}) \), you apply the even property: \( \cos(-45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \). This neat trick sidesteps the need to compute for negative angles directly.