PRACTICE PROBLEMS IN TRIGONOMETRY FOR GRADE 10
Practice Problems in Trigonometry for Grade 10 :
Here we are going to see some practice problems in the topic trigonometry.
Practice Problems in Trigonometry for Grade 10 - Quetsions
(1) Determine whether each of the following is an identity or not.
(i) cos²θ + sec²θ = 2 + sin θ
(ii) cot²θ + cos θ = sin² θ
(2) Prove the following identities
(i) sec²θ + cosec²θ = sec²θ cosec²θ
Question 1 :
Determine whether each of the following is an identity or not.
(i) cos²θ + sec²θ = 2 + sin θ
Solution :
To check whether the given question is an identity or not,first we have to simplify the left side question. So that we have to get the right side answer.
L.H.S
= cos²θ + sec²θ
= 1 - sin²θ + 1 + tan²θ
= 2 + tan²θ - sin²θ
Since it is not equal to R.H.S, it is not an identity.
(ii) cot²θ + cos θ = sin² θ
Solution :
L.H.S
= cot²θ + cos θ
= cosec²θ - 1 + cos θ
Since it is not equal to R.H.S, it is not an identity.
Question 2 :
Prove the following identities
(i) sec²θ + cosec²θ = sec²θ cosec²θ
Solution :
L.H.S
= sec²θ + cosec²θ
= (1/cos²θ) + (1/sin²θ)
taking L.C.M
= (sin²θ + cos²θ)/(cos²θ sin²θ)
value of sin²θ + cos²θ = 1
= 1/(cos²θ sin²θ)
= (1/cos²θ)(1/sin²θ)
= sec²θ cosec²θ
(ii) sin θ /(1-cos θ) = cosec θ + cot θ
Solution :
L.H.S
= sin θ /(1-cos θ)
Multiplying by the conjugate of denominator we get,
= [sin θ /(1-cos θ)] x [(1+cos θ)/(1+cos θ)]
instead of (1 + cos θ)/(1 + cos θ) we can write 1 - cos²θ by using the algebraic formula.
= [sin θ (1 + cosθ)]/(1-cos²θ)
= [sin θ (1 + cosθ)]/sin²θ
= (1 + cosθ)]/sinθ
= (1/sinθ) + (cosθ/sinθ)
= cosec θ + cot θ
(iii) √(1 - sin θ)/(1 + sin θ) = sec θ - tan θ
Solution :
L.H.S
= √(1-sin θ)/(1+sin θ)
= √(1-sin θ)/(1+sin θ) x (1-sin θ)/(1-sin θ)
= √(1-sin θ)²/[(1+sin θ) x (1-sin θ)]
= √(1-sin θ)²/(1²- sin²θ)
= √(1-sin θ)²/(cos²θ)
= √[(1-sin θ)/(cosθ)]²
= [(1-sin θ)/(cosθ)]
= [(1/cosθ)-(sin θ/cosθ)]
= sec θ - tan θ
R.H.S
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