Question 10 of 50intermediate🔧 ApplyShort Answer2 marks

Find the values of other five trigonometric functions: cot x = 3/4, x lies in third quadrant.

Correct Answer

Given cot x = 3/4 and x lies in third quadrant.

Step 1: tan x = 1/cot x = 4/3

Step 2: Using sec²x = 1 + tan²x, we get sec²x = 1 + 16/9 = 25/9. Hence sec x = ±5/3. Since x lies in third quadrant, sec x = -5/3.

Step 3: cos x = 1/sec x = -3/5

Step 4: sin x = tan x × cos x = (4/3) × (-3/5) = -4/5 (negative in third quadrant)

Step 5: cosec x = 1/sin x = -5/4

Therefore, sin x = -4/5, cos x = -3/5, tan x = 4/3, sec x = -5/3, cosec x = -5/4.

Exercise: EXERCISE 3.2 | Q: 3 | (Page 15)
For More Understanding

Explanation

This question follows the same pattern as Example 7 in the textbook. The key steps are: finding tan x as reciprocal of cot x, using the identity sec²x = 1 + tan²x, determining the sign based on quadrant (third quadrant means both sin and cos are negative, so sec and cosec are also negative), and then deriving remaining functions. The student must remember that in the third quadrant, only tan and cot are positive.

Solution Steps

  1. Step 1: Find tan x = 1/cot x = 4/3

  2. Step 2: Use identity sec²x = 1 + tan²x to get sec x = -5/3 (negative in third quadrant)

  3. Step 3: Find cos x = 1/sec x = -3/5

  4. Step 4: Find sin x = tan x × cos x = -4/5

  5. Step 5: Find cosec x = 1/sin x = -5/4

PreviousNext