Question 8 of 50intermediate🔧 ApplyShort Answer2 marks

Find the values of other five trigonometric functions: cos x = -1/2, x lies in third quadrant.

Correct Answer

Given cos x = -1/2 and x lies in third quadrant.

Step 1: Find sec x sec x = 1/cos x = 1/(-1/2) = -2

Step 2: Find sin x using identity sin²x + cos²x = 1 sin²x = 1 - cos²x = 1 - (-1/2)² = 1 - 1/4 = 3/4 sin x = ±3\sqrt{3}/2 Since x lies in third quadrant, sin x is negative. Therefore, sin x = -3\sqrt{3}/2

Step 3: Find cosec x cosec x = 1/sin x = -2/3\sqrt{3}

Step 4: Find tan x tan x = sin x/cos x = (-3\sqrt{3}/2)/(-1/2) = 3\sqrt{3}

Step 5: Find cot x cot x = 1/tan x = 1/3\sqrt{3}

Exercise: EXERCISE 3.2 | Q: 1 | (Page 15)
For More Understanding

Explanation

This question follows the exact pattern of Example 6 from the textbook. The key steps are: (1) Use reciprocal identity to find sec x, (2) Use Pythagorean identity sin²x + cos²x = 1 to find sin x, (3) Apply the quadrant rule - in the third quadrant, both sin x and cos x are negative, while tan x and cot x are positive. The student must correctly determine the sign of sin x based on the quadrant information.

Solution Steps

  1. Step 1: Find sec x using reciprocal identity

  2. Step 2: Use sin²x + cos²x = 1 to find sin x

  3. Step 3: Apply quadrant rule - in third quadrant sin is negative

  4. Step 4: Find cosec x as reciprocal of sin x

  5. Step 5: Find tan x = sin x/cos x

  6. Step 6: Find cot x as reciprocal of tan x

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