In Worksheet on finding the unknown angle using Trigonometric identities we will solve various types of practice questions on solving equation.
Here you will get 11 different types of solving equation using trigonometric identities questions with some selected questions hints.
1. Solve the equation 2 cos^{2} θ = 1; where 0° < θ < 90°.
2. Solve the equation \(\sqrt{3}\) tan θ = 1; where 0° < θ < 90°.
3. Solve the equation 2 cos (2θ) = 1; where 0° < θ < 90°.
4. Solve the equation 2 cos^{2} θ - 11 cos θ + 5 = 0; where 0° < θ < 90°.
5. Solve the equation 2 sin^{2} θ + cos θ - 2 = 0; where 0° < θ < 90°.
6. Solve the equation 2 sec^{2} θ = 5 tan θ; where 0° < θ < 90°.
Hint: 2 sec^{2} θ = 5 tan θ
⟹ 2 (1 + tan^{2} θ) = 5 tan θ
⟹ 2 + 2 tan^{2} θ = 5 tan θ
⟹ 2 tan^{2} θ - 5 tan θ + 2 = 0
⟹ 2 tan^{2} θ - 4 tan θ - tan θ + 2 = 0
⟹ 2 tan θ (tan θ – 2) – 1 (tan θ – 2) = 0
⟹ (tan θ – 2)(2 tan θ – 1) = 0
7. Are the following identities? If not, solve for θ, where 0°
< θ < 90°).
(i) \(\frac{ cos θ}{1 - sin θ}\) + \(\frac{ cos θ}{1 + sin θ}\) = 4
(i) \(\frac{ cos θ}{csc θ + 1}\) + \(\frac{ cos θ}{csc θ - 1}\) = 2.
8. Solve: 2 sin A – 1 = 0. Hence, prove that sin (3A) = 3 sin A – 4 sin^{3} A.
9. If 2 cos θ - \(\sqrt{3}\) = 0, find the value of θ.
Also, prove that sin (2θ) = 2 sin θ ∙ cos θ.
10. If 9 sin θ + 40 cos θ = 41, prove that 41 cos θ = 40.
11. If tan θ + sec θ = 3, where θ is acute then prove that 5 sin θ = 4.
Hint: tan θ + sec θ = 3
⟹ \(\frac{sin θ}{cos θ}\) + \(\frac{1}{cos θ}\) = 3
⟹ \(\frac{sin θ + 1}{cos θ}\) = 3
⟹ sin θ + 1 = 3 cos θ
⟹ (sin θ + 1)^{2} = (3 cos θ)^{2}
⟹ (sin θ + 1)^{2} = 9 cos^{2} θ
⟹ (sin θ + 1)^{2} = 9(1 - sin^{2} θ)
⟹ sin^{2} θ + 2 sin θ + 1 = 9 – 9 sin^{2} θ
⟹ 10 sin^{2} θ + 2 sin θ – 8 = 0
⟹ 5 sin^{2} θ + sin θ – 4 = 0
⟹ 5 sin^{2} θ + 5 sin θ - 4 sin θ – 4 = 0
⟹ 5 sin θ (sin θ + 1) - 4 (sin θ + 1) = 0
⟹ (sin θ + 1) (5 sin θ – 4) = 0
Answers on Worksheet on finding the unknown angle using trigonometric identities are given below to check the exact answers of the questions.
Answers
1. 60°
2. 30°
3. 30°
4. 60°
5. 60°
6. γ°, where tan γ° = ½ or 2.
7. (i) No; 60°
(ii) No; 45°
8. 30°
9. 30°
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