10th Maths 8.2
Chapter 8
Introduction to Trigonometry
NCERT Class 10th solution of Exercise 8.1
Trigonometric Ratios of Some Specific Angles
Exercise 8.2
Q1. Evaluate the following:
i) sin60∘cos30∘+sin30∘cos60∘
Sol. :
=sin60∘cos30∘+sin30∘cos60∘
=√32×√32+12×12
=34+14
=44
=1
Answer:
=1
ii) 2tan245∘+cos230∘-sin260∘
Sol. :
=2tan245∘+cos230∘-sin260∘
=2×(1)2+(√32)2-(√32)2
=2+34-34
=2
Answer:
=2
iii) cos45∘sec30∘+cosec30∘
Sol. :
=cos45∘sec30∘+cosec30∘
=1√22√3+2
=1√22+2√3√3
=1√2×√32+2√3
=√32√2+2√6
=√32(√6+√2)×√6-√2√6-√2
=√3(√6-√2)2(6-2)
=3√2-√68
Answer:
=3√2-√68
iv) sin30∘+tan45∘-cosec60∘sec30∘+cos60∘+cot45∘
Sol. :
=sin30∘+tan45∘-cosec60∘sec30∘+cos60∘+cot45∘
=12+1-2√32√3+12+1
=√3+2√3-42√34+√3+2√32√3
=3√3-43√3+4
=3√3-43√3+4×3√3-43√3-4
=(3√3-4)2(3√3+4)(3√3-4)
=27+16-24√327-16
=43-24√311
Answer:
=43-24√311
v) 5cos260∘+4sec230∘-tan245∘sin230∘+cos230∘
Sol. :
=5cos260∘+4sec230∘-tan245∘sin230∘+cos230∘
=5×(12)2+4×(2√3)2-(1)2(12)2+(√32)2
=54+163-114+34
=15+64-12121
=79-1212
=6712
Answer:
=6712
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Q2. Choose the correct option and justify your choice:
i) 2tan30∘1+tan230∘=
A) sin60∘ B) cos60∘ C) tan60∘ D) sin30∘
Sol. :
=2tan30∘1+tan230∘
=2×1√31+(1√3)2
=2√31+13
=2√343
=2√3×34
=√32
=sin60∘
Answer:
(A) sin60∘
ii) 1-tan245∘1+tan245∘=
A) tan90∘ B) 1 C) sin45∘ D) 0
Sol. :
=1-tan245∘1+tan245∘
=1-(1)21+(1)2
=1-11+1
=02
=0
Answer:
(D) 0
iii) sin2A=2sinA is true when A=
A) 0∘ B) 30∘ C) 45∘ D) 60∘
Sol. :
sin2A=2sinA
sin(2×0∘)=2sin0
sin0=2sin0
0=2×0
=0
Answer:
(A) 0
iv) 2tan30∘1-tan230∘=
A) cos60∘ B) sin60∘ C) tan60∘ D) sin30∘
Sol. :
=2tan30∘1-tan230∘
=2×1√31-(1√3)∘
=2√31-13
=2√323
=2√3×32
=√3
=tan60∘
Answer:
(C) tan60∘
Q3. If tan(A+B)=√3 and tan(A-B)=1√3;0∘<A+B≤90∘;A<B, find A and B.
Sol. :
tan(A+B)=√3
tan(A+B)=tan60∘
A+B=60∘_________(1)
and
tan(A-B)=1√3
tan(A-B)=tan30∘
A-B=30∘_________(2)
We add equations (1) and (2)
A+B+A-B=60∘+30∘
2A=90∘
A=90∘2
A=45∘
this value is put in equation (1)
45∘+B=60∘
B=60∘-45∘
B=15∘
Answer:
A=45∘,B=15∘
Q4. 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 value of θ.
v) cotA is not defined for A=0∘.
Answer:
i) False ii) True iii) False iv) False v) False.
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