# How do you evaluate ?

Samantha

Solution:

Step:1 For any right triangle, there are 6 trigonometric ratios: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec) and cotangent (cot).

Trigonometric Ratios

Consider a triangle ABC, right angled at B

Step:2 In the above right triangle ABC, AB is base (b), BC is perpendicular (p) and AC is hypotenuse (h).

Sin is defined as the ratio of perpendicular to hypotenuse, cos is defined as the ratio of base to hypotenuse and tan is defined as the ratio of perpendicular to base. There exists three more ratios which are reciprocal of sin, cos and tan.

For angle A, these ratios can be listed as shown below.

tan A is the ratio of sin A and cos A, which can be written as .

Cosec A is reciprocal of sin A sec A is reciprocal of cos A.

Trigonometric ratios of some basic angles as  are defined.

For some other angles trigonometric ratios can be derived from these ratios.

How to memorize all the trigonometric ratios?

Step:3 There is a table- trick to memorize all these ratios for the angles written above.

Step:4 In the row of “sin” put the numbers from 0 to 4 in increasing order and then in the row of “cos” put all the numbers from 4 to 0 in decreasing order.

Divide all the numbers by 4.

Write all the values in square root.

Step:5 All the values of table can be simplified and written. The value of tan is ratio of sin and cos. Cosec is reciprocal of sin, sec is reciprocal of cos and cot is reciprocal of tan.

The completed table is made hereunder.

To measure the angles two units are used namely radian and degree.

radian can be converted into degree using the above conversion.

Now to find the value of , evaluate the value of .

From Table 4 it can be seen clearly that the value of is .

Explanation of .

Consider a right triangle with an angle of .

Now sin is defined as the ratio of perpendicular and hypotenuse.

From the Figure.1, the value of  is calculated as follows:

Hence, the value of  that is  can be find using the table or diagram.

Similar Problems:

Problem 1. How do you find the value of ?

Solution.

Step:1 To measure the angles two units are used namely radian and degree.

radian can be converted into degree using the above conversion.

Step:2 Now to find the value of , evaluate the value of .

From Table 4 it can be seen clearly that the value of is .

Step:3 From the above Figure.1, the value of  is calculated as follows:

Answer. The value of is .

Problem 2. How do you find the value of ?

Solution.

Step:1 To measure the angles two units are used namely radian and degree.

radian can be converted into degree using the above conversion.

Step:2 Now to find the value of , evaluate the value of .

From Table 4 it can be seen clearly that the value of  is 2.

Step:3 From the above Figure.1, the value of  is calculated as follows:

Answer. The value of is .

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