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  =  sin²θ

(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 + sinθ

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  =  sin²θ

Solution :

L.H.S

  =  cot²θ + cosθ

  =  cosec²θ - 1 + cosθ

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)

=  (sin²θ + cos²θ)/(cos²θ ⋅ sin²θ)

Value of sin²θ + cos²θ  =  1.

Then, 

  =  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)] ⋅ [(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) ⋅ (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

After having gone through the stuff above, we hope that students would have understood how to solve word problems in trigonometry. 

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