In Δ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine:

(1) sin A, cos A                                   

(2) sin C, cos C

 

In Fig. 8.13, find tan P – cot R.

If sin A = , calculate cos A and tan A.

Given 15 cot A = 8, find sin A and sec A.

Given sec θ = , calculate all other trigonometric ratios.

If ∠ A and ∠ B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B.

If cot θ =   evaluate:

(1) {(1 + sin θ) (1 - sin θ)} ÷ {(1 + cos θ) (1 - cos θ)}                         

(2) cot² θ

If 3 cot A = 4, check whether (1 - tan² A) ÷ (1 + tan² A) = cos² A – sin² A or not.

 

In triangle ABC, right-angled at B, if tan A = , find the value of:

(1) sin A cos C + cos A sin C                                 

(2) cos A cos C – sin A sin C        

 

In Δ PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm.

Determine the values of sin P, cos P and tan P.

State whether the following are true or false. Justify your answer.

(1) The value of tan A is always less than 1.

(2) sec A =  for some value of angle A.

(3) cos A is the abbreviation used for the cosecant of angle A.

(4) cot A is the product of cot and A.

(5) sin θ =  for some angle θ.

Evaluate the following:

(1) sin 60⁰ cos 30⁰ + sin 30⁰ cos 60⁰                      

(2) 2 tan² 45⁰ + cos² 30⁰ – sin² 60⁰

(3) cos 45⁰ ÷ (sec 30⁰ + cosec 30⁰)   

(4) (sin 30⁰ + tan 45⁰ – cosec 60⁰) ÷ (sec 30⁰ + cos 60⁰ + cot 45⁰)

(5) (5 cos² 60⁰ + 4 sec² 30⁰ - tan² 45⁰) ÷ (sin² 30⁰ + cos² 30⁰)

Choose the correct option and justify your choice:

(1) 2 tan 30⁰ ÷ (1 + tan² 30⁰) =

(A) sin 60⁰                    

(B) cos 60⁰                          

(C) tan 60⁰                       

(D) sin 30⁰

(2) (1 – tan² 45⁰) ÷ (1 + tan² 45⁰) =

(A) tan 90⁰                            

(B) 1                             

(C) sin 45⁰                              

(D) 0

(3) sin 2A = 2 sin A is true when A =

(A) 0⁰                                     

(B) 30⁰                         

(C) 45⁰                                     

(D) 60⁰

(4) 2 tan 30⁰ ÷ (1 – tan² 30⁰) =

(A) cos 60⁰                            

(B) sin 60⁰                   

(C) tan 60⁰                              

(D) sin 30⁰

If tan (A + B) = √3 and tan (A – B) = ; 0⁰ < A + B ≤ 90⁰; A > B, find A and B.

 

State whether the following are true or false. Justify your answer.

(1) sin (A + B) = sin A + sin B.

(2) The value of sin θ increases as θ increases.

(3) The value of cos θ increases as θ increases.

(4) sin θ = cos θ for all values of θ.

(5) cot A is not defined for A = 0⁰.

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

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

Choose the correct option. Justify your choice.

(1) 9 sec² A – 9 tan² A =

(A) 1                     

(B) 9                   

(C) 8                                 

(D) 0

(2) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =

(A) 0                     

(B) 1                   

(C) 2                                 

(D) -1

(3) (sec A + tan A) (1 – sin A) =

(A) sec A              

(B) sin A             

(C) cosec A                     

(D) cos A

(4) (1 + tan² A) ÷ (1 + cot² A) =

(A) sec² A            

(B) –1                 

(C) cot² A                       

(D) tan² A

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(1) (cosec θ – cot θ)² =         

(2)  +  = 2 sec A

(3)   +   = 1 + sec θ + cosec θ

[Hint: Write the expression in terms of sin θ and cos θ]

(4)   = sin² A ÷ (1 – cos A)

[Hint: Simplify LHS and RHS separately]

(5) (cos A – sin A + 1) ÷ (cos A + sin A - 1) = cosec A, using the identity cosec² A = 1 + cot² A.

(6) √ {(1 + sin A) ÷ (1 – sin A)} = sec A + tan A

(7) (sin θ – 2 sin³ θ ÷ (2cos³ θ – cos θ) = tan θ

(8) (sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

(9) (cosec A – sin A) (sec A – cos A) = 

[Hint: Simplify LHS and RHS separately]

(10) (1 + tan² A) ÷ (1 + cot² A) = (1 - tan² A) ÷ (1 - cot² A) = tan² A