Question

Evaluate the function without using a calculator. \(\cos \frac{7 \pi}{4}\)

Short answer

The value of \( \cos \frac{7 \pi}{4} \) is \( \sqrt2/2 \)

Step-by-step solution

Understand the concept of the unit circle

A unit circle is a circle of radius one. The trigonometric functions of angles can be defined in terms of the coordinates of points on the unit circle. In a unit circle, the cos of an angle gives us the x-coordinate of the point on the circle formed by that angle. The angle is always measured counter-clockwise, starting from the x-axis.

Understand the relative angles

The circle has 2π radians (or 360 degrees), so the angle \(7\pi/4\) is equivalent to \(2\pi - \pi/4\) which is exactly the same as \(- \pi/4\). This represents -45 degrees, and the cosine of any angle is equal to the cosine of its negative. Hence, \( \cos \frac{7 \pi}{4} = \cos ( -\frac{\pi}{4} ) \)

Evaluate cosine on the unit circle

Referring to the unit circle, we know that the cosine of 45 degrees (or \(\pi/4\) radians) is \(\sqrt2/2\). Thus, the cosine of multiplies of 45 degrees is also \(\sqrt2/2\), for this problem, \( \cos \frac{7 \pi}{4} = \sqrt2/2 \)

Key concepts

Trigonometric Functions

Trigonometric functions form the basis for trigonometry, which helps us understand the relationships between the angles and sides of triangles. They are also crucial in analyzing periodic phenomena in various fields such as physics and engineering. The primary trigonometric functions are sine, cosine, and tangent.

• Sine (sin): In a unit circle, it represents the y-coordinate of an angle.
• Cosine (cos): Represents the x-coordinate of an angle on the unit circle.
• Tangent (tan): Represents the ratio of sine to cosine for a given angle, or the slope of the radius at that angle.

The unit circle, a circle with a radius of one, is very handy for defining these functions. By positioning an angle with its vertex at the circle's center, the sine and cosine give us the vertical and horizontal distances from the center to the edge of the circle, respectively. Understanding these relationships allows us to solve various real-world problems about cycles, oscillations, and waves. These concepts also help us find exact values of angles' trigonometric functions without a calculator, just like in the given problem.

Cosine Function

The cosine function is one of the most crucial elements of trigonometry. It maps an angle in a right triangle, or from the unit circle, to an x-coordinate. This function is crucial in determining the horizontal distance from the center of a unit circle to a point on its circumference reached by the angle.

• The function is defined as \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\).
• On the unit circle, it gives the horizontal coordinate of a point formed by rotating from the positive x-axis by the angle \(\theta\).
• Cosine values repeat every full circle (360 degrees or \(2\pi\) radians), making it a periodic function.

In trigonometric problem-solving, finding the cosine of specific angles becomes essential. In our exercise, understanding that the cosine of \(\frac{7\pi}{4}\) can derive via the unit circle and angle properties without using a calculator showcases its significance in precise calculations without electronic aid.

Radian Measure

Radian measure is an alternative to degrees for measuring angles. It is a way to express angles as fractions of a full circle. Instead of dividing a circle into 360 parts (degrees), it considers an angle as a ratio of the arc length to the radius of a circle.

• One full circle is measured as \(2\pi\) radians.
• An angle of \(\pi\) radians is equivalent to half a circle, or 180 degrees.
• Smaller angle measures include \(\frac{\pi}{4}\) radians which correspond to 45 degrees.

Using radians is often preferred in calculus and higher mathematics for a few reasons. It simplifies the formulas involving periodic functions, such as sine and cosine. Additionally, radians relate more naturally to arc length and their related calculations, which is especially useful in sciences and engineering. In our example, understanding how to express \(\frac{7\pi}{4}\) in radians and convert it to a known angle helps us find the cosine without needing conversion to degrees.

Angle Conversion

Angles can be represented in different units, primarily degrees and radians. Converting between these two measurements is a common task in trigonometry and other branches of mathematics.

• From degrees to radians: Use the formula \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\).
• From radians to degrees: Use the formula \(\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\).

Understanding angle conversion helps in visualizing angles in familiar terms, especially when dealing with trigonometric functions like sine and cosine. For instance, converting \(\frac{7\pi}{4}\) radians to degrees can make it easier to link with common angle measures. Our exercise highlights this by recognizing that \(\frac{7\pi}{4}\) is equivalent to \(-\frac{\pi}{4}\), which is an angle often memorized in degrees as -45 degrees. Recognizing these equivalents can make evaluating trigonometric functions much simpler.