1. In , right-angled at B , . Determine: (i) sinA , cosA (ii) sinC , cosC
Solution : Here , AB=24 cm , BC=7 cm
In∆ABC, we have
(i) and
(ii) and
2. In Fig. 8.13, find .
Solution : Here, PQ=12 cm , PR=13 cm
In∆PQR , we have
Now ,
and
3. If , calculate cosA and tanA .
Solution : Given
let
In∆ABC , we have
and
4. Given , find sinA and secA .
Solution : Given,
Let,
In∆ABC , we have
and
5. Given , calculate all other trigonometric ratios .
Solution : Given,
Let
In∆ABC , we have
and
6. If and are acute angles such that , then show that .
Solution : Given,
So, ABC is an isosceles triangle .
proved .
7. If , evaluate : (i) (ii)
Solution: Given,
Let,
In∆ABC , we have
and
(i) We have,
(ii) We have,
8. If , check whether or not .
Solution: Given ,
Let,
In∆ABC , we have
LHS:
RHS :
9. In triangle ABC, right-angled at B , if , find the value of : (i) (ii)
Solution : Given,
Let
In∆ABC , we have
(i) We have,
(ii) We have,
10. In , right-angled at Q, and . Determine the values of and .
Solution : Here ,
and
In , we have
11. State whether the following are true or false . Justify your answers :
(i) The value of tanA is always less than 1 .
(ii) for some value of angle A.
(iii) cosA is the abbreviation used for the cosecant of angle A .
(iv) cotA is the product of cot and A .
(v) for some angles θ .
Solution: (i) False ,because the value of tanA is not always less than 1. For example : .
(ii) True , because the hypotenuse is the longest side in a right triangle,
then the value of secA is always greater than or equal to 1 .
(iii) False, because cosA is the abbreviation used for the cosine of angle A .
(iv) False , because cotA is the ratio of base and perpendicular of the right triangle .
(v) False , because the hypotenuse is the longest side in a right triangle, then the value of sinθ is always less than 1 .
1. Evaluate the following : (i) (ii) (iii) (iv) (v)
Solution : (i) We have,
(ii) We have,
(iii) We have,
(iv) We have,
(v) We have,
2. Choose the correct option and justify your choice : (i)
(A) (B) (C) (D)
Solution: (A)
[Hint: We have,
]
(ii)
(A) (B) 1 (C) (D) 0
Solution: (D) 0
[ Hint: We have,:
]
(iii) is true when
(A) 0° (B) 30° (C) 45° (D) 60°
Solution : (A) 0°
[ Hint: Putting
LHS :
RHS : ]
(iv)
(A) (B) (C) (D)
Solution : (C)
[ Hint: We have,
]
3. If and ,find and .
Solution : We have,
and
Putting in equation , we have
Therefore , the value of A and B are 45° and 15° respectively .
4. State whether the following are true or false . Justify your answer .
(i) sin(A+B) = sinA+sinB
(ii) The value of sinθ increases as θ increases .
(iii) The value of cosθ increases as θ increases .
(iv) sinθ = cosθ for all values of θ .
(v) cotA is not defined for A = 0°
Solution : (i) sin(A + B) = sinA + sinB
False , Because if you are putting the value of A and B then both sides are not equal .
(ii) The value of sinθ increases as θ increases .
(ii) True , because the value of sinθ increases asθ increases .
(iii) False , because the value of cosθ increases as θ increases .
[ i.e., the value of cosθ increases as θ decreases .]
(iv) False , only for θ=45° is equal and other value of θ is not equal both sides .
(v) True, because the value of A=0° is not defined for cotA .
1. Evaluate : (i) (ii) (iii) (iv)
Solution : (i) We have,
(ii) We have ,
(iii) We have,
(iv) We have,
2. Show that :
(i)
(ii)
Solution :
(i) LHS :
RHS
(ii) LHS :
RHS
3. If where is an acute angle , find the value of A .
Solution: We have,
Since 90° – 2A and A – 18° are both acute angles .
Therefore, the value A is 36° .
4. If tanA = cotB , prove that .
Solution: Given,
Since A and are both acute angles .
Proved .
5. If where 4A is an acute angle, find the value of A .
Solution : We have,
Since and are both acute angles .
Therefore, the value A is 22° .
6. If A , B and C are interior angles of a triangle ABC , then show that .
Solution : Since, A , B and C are interior angles of a triangle ABC respectively .
7. Express in terms of trigonometric ratios of angles between 0° and 45° .
Solution : We have ,
1. Express the trigonometric ratios sinA , secA and tanA in terms of cotA .
Solution : We know that ,
and
2. Write all the other trigonometric ratios of in terms of secA .
Solution : We have ,
and
3. Evaluate : (i) (ii)
Solution: (i) We have ,
(ii) We have,
4. Choose the correct option . Justify your choice .
(i)
(A) 1 (B) 9 (C) 8 (D) 0
Solution: (B) 9
[ hints :
]
(ii)
(A) 0 (B) 1 (C) 2 (D)
Solution: (C) 2
[Hints: We have ,
]
(iii)
(A) (B) (C) (D)
Solution: (D) cosA
[ Hints: We have,
]
(iv)
(A) (B) (C) (D)
Solution: (D)
[ Hints: We have ,
]
5. Prove the following identities , where the angles involved are acute angles for which the expressions are defined .
(i)
Solution : (i) L.H.S :
= R.H.S Proved.
(ii)
Solution: L.H.S :
= R.H.S. Proved.
(iii)
[Hint : Write the expression in terms of sinθ and cosθ]
Solution : L.H.S :
= R.H.S. Proved
(iv)
[Hint : Simplify LHS and RHS separately]
Solution: L.H.S :
RHS :
LHS = RHS Proved
(v) , using the identity .
Solution: LHS :
RHS Proved.
(vi)
Solution : LHS :
= RHS Proved
(vii)
Solution : L.H.S. :
= R.H.S. Proved .
(viii)
Solution : L.H.S. :
R.H.S Proved.
(ix)
[ Hint :Simplify LHS and RHS separately ]
Solution: LHS :
RHS :
LHS = RHS Proved .
(x)
Solution: First part :
Second part :
First part = Second part = Third part . Proved.