How do you evaluate ?

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Answer


Celeste

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.

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:4 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.

                                                                             

 

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

Explanation of :

consider a right triangle with an angle of .

Now cos is defined as the ratio of base and hypotenuse.

From Figure.1 value of  can be calculated as follows:

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

 

 

 

Similar Problems:

Problem 1. What is the value of ?

Solution.

Step:1 From the Table 4, it is seen that the value of  is 0.

Step:2 From the above Figure.1, the value of  is calculated as follows:

 

Answer. The value of is .

 

 

Problem 2. What is the value of ?

Solution.

Step:1 From the Table 4, it is seen that the value of is .

Step:2 From the above Figure.1, the value of  is calculated as follows:

 

Answer. The value of is .

 

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