Question 11 of 50intermediate🔧 ApplyShort Answer2 marks

Find the values of other five trigonometric functions: sec x = 13/5, x lies in fourth quadrant.

Correct Answer

Given sec x = 13/5 and x lies in fourth quadrant.

Step 1: Since sec x = 1/cos x, we have cos x = 5/13.

Step 2: Using the identity sin²x + cos²x = 1, we get sin²x = 1 - cos²x = 1 - 25/169 = 144/169. Hence sin x = ±12/13. Since x lies in fourth quadrant where sin x is negative, sin x = -12/13.

Step 3: Therefore, cosec x = 1/sin x = -13/12.

Step 4: Now tan x = sin x/cos x = (-12/13)/(5/13) = -12/5.

Step 5: Hence cot x = 1/tan x = -5/12.

The five trigonometric functions are: cos x = 5/13, sin x = -12/13, cosec x = -13/12, tan x = -12/5, cot x = -5/12.

Exercise: EXERCISE 3.2 | Q: 4 | (Page 15)
For More Understanding

Explanation

This question follows the same pattern as Example 7 in the textbook. The key is to use the fundamental identity sin²x + cos²x = 1 and remember the sign conventions for each quadrant. In the fourth quadrant, cos x and sec x are positive, while sin x, cosec x, tan x, and cot x are negative.

Solution Steps

  1. Step 1: Find cos x = 1/sec x = 5/13

  2. Step 2: Use sin²x + cos²x = 1 to get sin x = -12/13 (negative in IV quadrant)

  3. Step 3: Find cosec x = -13/12

  4. Step 4: Find tan x = sin x/cos x = -12/5

  5. Step 5: Find cot x = -5/12

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