# How to Find Exact Value of Trig Functions Without Calculator

Trigonometric functions are essential in various fields of study, such as mathematics, physics, engineering, and many more. These functions include sine, cosine, tangent, cotangent, secant, and cosecant. They are used to calculate the relationships between angles and sides of a right-angled triangle.

Normally, to find the exact value of these functions, one needs a scientific calculator. However, what if you don’t have a calculator and still need to find the answer? This article will guide you on how to find the exact values of trigonometric functions without the help of a calculator.

## The Unit Circle

To find the exact values of trigonometric functions, one must understand the unit circle. A unit circle is a circle with a radius of one unit. In trigonometry, we use the unit circle to determine the values of trigonometric functions for any angle measure.

The unit circle is divided into four quadrants: Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. Each quadrant has a unique set of coordinates. Quadrant I represents values for angles between 0 and 90 degrees, Quadrant II represents angles between 90 and 180 degrees, Quadrant III represents angles between 180 and 270 degrees, and Quadrant IV represents values for angles between 270 and 360 degrees.

### Sine Function

The sine function is used to calculate the ratio of the opposite side of a right triangle to its hypotenuse. In the unit circle, the sine function is determined by finding the y-coordinate of a point on the circle. For example, if we have an angle of 60 degrees, we can find the value of sin 60 by locating the point on the unit circle that intersects with the angle of 60 degrees. The y-coordinate of the point is equal to sin 60.

To determine the value of sin 60:

Step 1: Draw a unit circle and locate the angle of 60 degrees in Quadrant I.

Step 2: Draw a line from the origin to the point where the angle intersects with the circle.

Step 3: Determine the y-coordinate of the point where the line intersects with the circle.

Step 4: The y-coordinate is equal to the value of sin 60, which is 0.866.

### Cosine Function

The cosine function is used to calculate the ratio of the adjacent side of a right triangle to its hypotenuse. In the unit circle, the cosine function is determined by finding the x-coordinate of a point on the circle. For example, if we have an angle of 45 degrees, we can find the value of cos 45 by locating the point on the unit circle that intersects with the angle of 45 degrees. The x-coordinate of the point is equal to cos 45.

To determine the value of cos 45:

Step 1: Draw a unit circle and locate the angle of 45 degrees in Quadrant I.

Step 2: Draw a line from the origin to the point where the angle intersects with the circle.

Step 3: Determine the x-coordinate of the point where the line intersects with the circle.

Step 4: The x-coordinate is equal to the value of cos 45, which is 0.707.

### Tan Function

The tan function is used to calculate the ratio of the opposite side of a right triangle to its adjacent side. In the unit circle, the tan function is determined by finding the y-coordinate divided by the x-coordinate of a point on the circle. For example, if we have an angle of 30 degrees, we can find the value of tan 30 by locating the point on the unit circle that intersects with the angle of 30 degrees. The y-coordinate of the point divided by the x-coordinate is equal to tan 30.

To determine the value of tan 30:

Step 1: Draw a unit circle and locate the angle of 30 degrees in Quadrant I.

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Step 2: Draw a line from the origin to the point where the angle intersects with the circle.

Step 3: Determine the y-coordinate and the x-coordinate of the point where the line intersects with the circle.

Step 4: Divide the y-coordinate by the x-coordinate to determine the value of tan 30, which is 0.577.

### Cot Function

The cot function is used to calculate the ratio of the adjacent side of a right triangle to its opposite side. In the unit circle, the cot function is determined by finding the x-coordinate divided by the y-coordinate of a point on the circle. For example, if we have an angle of 60 degrees, we can find the value of cot 60 by locating the point on the unit circle that intersects with the angle of 60 degrees. The x-coordinate of the point divided by the y-coordinate is equal to cot 60.

To determine the value of cot 60:

Step 1: Draw a unit circle and locate the angle of 60 degrees in Quadrant I.

Step 2: Draw a line from the origin to the point where the angle intersects with the circle.

Step 3: Determine the x-coordinate and the y-coordinate of the point where the line intersects with the circle.

Step 4: Divide the x-coordinate by the y-coordinate to determine the value of cot 60, which is 1.732.

### Sec Function

The sec function is used to calculate the ratio of the hypotenuse of a right triangle to its adjacent side. In the unit circle, the sec function is determined by finding the reciprocal of the x-coordinate of a point on the circle. For example, if we have an angle of 75 degrees, we can find the value of sec 75 by locating the point on the unit circle that intersects with the angle of 75 degrees. The reciprocal of the x-coordinate of the point is equal to sec 75.

To determine the value of sec 75:

Step 1: Draw a unit circle and locate the angle of 75 degrees in Quadrant II.

Step 2: Draw a line from the origin to the point where the angle intersects with the circle.

Step 3: Determine the x-coordinate of the point where the line intersects with the circle.

Step 4: Find the reciprocal of the x-coordinate to determine the value of sec 75, which is 1.301.

### Cosec Function

The cosec function is used to calculate the ratio of the hypotenuse of a right triangle to its opposite side. In the unit circle, the cosec function is determined by finding the reciprocal of the y-coordinate of a point on the circle. For example, if we have an angle of 120 degrees, we can find the value of cosec 120 by locating the point on the unit circle that intersects with the angle of 120 degrees. The reciprocal of the y-coordinate of the point is equal to cosec 120.

To determine the value of cosec 120:

Step 1: Draw a unit circle and locate the angle of 120 degrees in Quadrant II.

Step 2: Draw a line from the origin to the point where the angle intersects with the circle.

Step 3: Determine the y-coordinate of the point where the line intersects with the circle.

Step 4: Find the reciprocal of the y-coordinate to determine the value of cosec 120, which is 1.155.

## FAQs

### 1. Can I find exact values of trigonometric functions without a calculator?

Yes, you can. By using the unit circle and the definitions of trigonometric functions, you can calculate the exact values of these functions for any angle measure.

### 2. Why are trigonometric functions important?

Trigonometric functions are crucial in various fields of study, such as mathematics, physics, engineering, and many more. They are used to calculate the relationships between angles and sides of a right-angled triangle, making them an essential tool in problem-solving and analysis.

### 3. Why is the unit circle important in trigonometry?

The unit circle is used to determine the values of trigonometric functions for any angle measure. It provides a convenient way to visualize all the possible values of sine, cosine, tangent, cotangent, secant, and cosecant at different angles.

### 4. What are the six trigonometric functions?

The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. They describe the relationships between the angles and sides of a right-angled triangle.

### 5. Why do we need to know the exact values of trigonometric functions?

Knowing the exact values of trigonometric functions is crucial in solving complex problems in mathematics, physics, engineering, and many more fields. It allows us to perform accurate calculations and make more precise measurements and predictions.