Question 16 of 18intermediate🔧 ApplyShort Answer2 marks

Write all the other trigonometric ratios of ∠A\angle A in terms of sec⁡A\sec A.

Correct Answer

We know that sec A is the reciprocal of cos A. Therefore:

cos A = 1/sec A

Using the identity sec² A = 1 + tan² A from the context, we get: tan A = sec2A–1\sqrt{sec² A – 1}

Since cot A is the reciprocal of tan A: cot A = 1/sec2A–1\sqrt{sec² A – 1}

Using the identity sin² A + cos² A = 1, we get: sin A = 1–cos2A\sqrt{1 – cos² A} = 1–1/sec2A\sqrt{1 – 1/sec² A} = sec2A–1\sqrt{sec² A – 1}/sec A

Since cosec A is the reciprocal of sin A: cosec A = sec A/sec2A–1\sqrt{sec² A – 1}

Exercise: EXERCISE 8.3 | Q: 2 | (Page Page 19)
For More Understanding

Explanation

The question requires expressing all five other trigonometric ratios in terms of sec A. The context provides the reciprocal relationships (sec A = 1/cos A, cosec A = 1/sin A, cot A = 1/tan A) and the identity sec² A = 1 + tan² A. Using these, we can derive all other ratios. The identity sin² A + cos² A = 1 is also referenced in the context through the Pythagoras theorem application.

Solution Steps

  1. Step 1: Express cos A as reciprocal of sec A: cos A = 1/sec A

  2. Step 2: Use identity sec² A = 1 + tan² A to find tan A = sec2A–1\sqrt{sec² A – 1}

  3. Step 3: Find cot A as reciprocal of tan A

  4. Step 4: Use sin² A = 1 – cos² A and substitute cos A to find sin A

  5. Step 5: Find cosec A as reciprocal of sin A

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