Trigonometry Most important questions

CHAPTER – 8 & 9
TRIGONOMETRY
Trigonometric Ratios (T - Ratios) of an acute angle of a right triangle
In XOY-plane, let a revolving line OP starting from OX, trace out XOP=From P (x, y)draw PM
 to OX.
In right angled triangle OMP. OM = x (Adjacent side); PM = y (opposite side); OP = r (hypotenuse).




Opposite Side y Adjacent Side x Opposite Side y
sin    , cos    , tan   
Hypotenuse r Hypotenuse r Adjacent Side x
Hypotenuse r Hypotenuse r Adjacent Side x
cos ec   , sec    , cot   
Opposite Side y Adjacent Side x Opposite Side y

Reciprocal Relations
1 1 1
cos ec  , sec   and cot  
sin  cos  tan 

Quotient Relations
sin  cos 
tan   and cot  
cos  sin 

IMPORTANT QUESTIONS
4
If tan A  , find the value of all T– ratios of θ .
3
BC 4
Solution: Given that, In right Δ ABC, tan A  
AB 3
Therefore, if BC = 4k, then AB = 3k, where k is a positive number.
Now, by using the Pythagoras Theorem, we have
AC2 = AB2 + BC2 = (4k)2 + (3k)2 = 25k2
So, AC = 5k
Now, we can write all the trigonometric ratios using their definitions.
BC 4k 4 AB 3k 3
sin A    , cos A   
AC 5k 5 AC 5k 5
1 3 1 5
and cot A   , cos ecA   ,
tan A 4 sin A 4
1 5
sec A  
cos A 3

, Questions for Practice
5
1. If sin θ  , find the value of all T– ratios of θ .
13
7
2. If cos θ  , find the value of all T– ratios of θ .
25
15
3. If tanθ  , find the value of all T– ratios of θ .
8
4. If cot θ  2 , find the value of all T– ratios of θ .
5. If cosec θ  10 , find the value of all T– ratios of θ .
6. In Δ OPQ, right-angled at P, OP = 7 cm and OQ – PQ = 1 cm. Determine the values of sin Q and
cos Q.
7. 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.



Trigonometric ratios for angle of measure.
00, 300,450, 600 and 900 in tabular form.

A 00 300 450 600 900
1 1 3
sinA 0 1
2 2 2
3 1 1
cosA 1 0
2 2 2
1
tanA 0 1 3 Not defined
3
2
cosecA Not defined 2 2 1
3
2
secA 1 2 2 Not defined
3
1
cotA Not defined 3 1 0
3

IMPORTANT QUESTIONS
3
If cos (A – B) = and sin (A + B) = 1, then find the value of A and B.
2
3
Solution: Given that cos( A  B )   cos 300
2
0
 A  B  30 ………………. (1)
and sin( A  B)  1  sin 900
 A  B  900 …………………… (2)
Solving equations (1) and (2), we get A = 600 and B = 300.

Questions for Practice
Evaluate each of the following:
1. sin 600 cos 300  cos 600 sin 300