Class 10 Maths Chapter 8: Introduction to Trigonometry Ex 8.1

Question 1
In Δ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i) Sin A, cos A
(ii) Sin C, cos C
Solution
In Δ ABC, right-angled at B ,using Pythagoras theorem we have
AC² = AB² +BC²  = 576 + 49 = 625
Or AC=25 ( taking positive value only)
Now
(i) In a right angle triangle ABC where B=90° ,

Sin A = BC/AC = 7/25
CosA = AB/AC = 24/25
(ii)

Sin C = AB/AC = 24/25
Cos C = BC/AC =7/25

Question 2
In below find tan P – cot R

Solution
Again consider the above figure . Now by Pythagoras theorem
PQ² + QR²  =PR²
QR=5
Now
tan P = Perp/Base = 5/12
Cot R = Base/Perm = 5/12
So tan P – cot R=0

Question 3. If $sin A= \frac{3}{4}$. Calculate cos A and tan A.
Solution
Given $sin A= \frac{3}{4}$
Or P/H=3/4
Let  P=3k and H=4k

Now By Pythagoras theorem
P² + B² =H²
9k² + B² =16k²
Or  $ B =+ k \sqrt 7 $
Now $\cos A = {B \over H} = {{\sqrt 7 } \over 4}$
Now $\tan A = {{\sin A} \over {\cos A}} = {3 \over {\sqrt 7 }}$
Question 4
Given 15 cot A = 8, find sin A and sec A.

Solution 4
$Cot A =\frac{8}{15}$
Or
${B \over P} = {8 \over {15}}$
Let B=8K and P=15k

So in a right angle triangle with angle A
P² + B² =H²
Or H=17K
Sin A = P/H = 15/17
Sec A = H/B = 17/8

Question 5
Given sec θ =13/12. Calculate all other trigonometric ratios.
Solution 5
Given sec θ=13/12
Or
H/B=13/12
let H=13K ,B=12K

So in a right angle triangle with angle A
P² + B² =H²
P=5k
sin θ = P/H = 5/13
Cos θ = B/H = 12/13
Tan θ = P/B = 5/12
Cosec θ= 1/sin θ = 13/5
cot θ = 1/tan θ = 12/5


Question 6
If ∠ A and ∠ B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B.
Solution 6
In a triangle
Cos A =cos B

AC/AB= BC/AB
⇒AC=BC
⇒Angle A and Angle B

Question 7
If cot θ =7/8 evaluate
(i) ${{\left( {1 + \sin\theta } \right)\left( {1 - \sin\theta } \right)} \over {\left( {1 + \cos\theta } \right)\left( {1 - \cos\theta } \right)}}$
(ii) ${\cot ^2}\theta $
Solution
Given
Given cot θ=7/8
Or
B/P=7/8
let B=7K ,P=8K

So in a right angle triangle with angle θ
P² + B² =H²
$H = k\sqrt {113} $
$\sin \theta  = {P \over H} = {8 \over {\sqrt {113} }}$
$\cos \theta  = {B \over H} = {7 \over {\sqrt {113} }}$
(i) ${{\left( {1 + sin\theta } \right)\left( {1 - sin\theta } \right)} \over {\left( {1 + cos\theta } \right)\left( {1 - cos\theta } \right)}}$
$ = {{1 - \sin^2 \theta } \over {1 - \cos^2 \theta }}$
${{1 - {{64} \over {113}}} \over {1 - {{49} \over {113}}}} = {{49} \over {64}}$
(ii)Cot² θ=(cot θ)²= 49/64

Question 8
If 3 cot A = 4, check whether below is true or not
${{1 - \tan^2 A} \over {1 + \tan^2 A}} = \cos^2 A - \sin^2 A$
Solution 8
Given
Cot A=4/3
Or
B/P=4/3
Let B=4k and P=3k

So in a right angle triangle with angle A
P² + B² =H²
H=5k
Now tan A=1/cot A=3/4
Cos A=B/H=4/5
Sin A=P/H=3/5
Let us take the LHS
${{1 - ta{n^2}A} \over {1 + ta{n^2}A}} = {{1 - {{\left( {{3 \over 4}} \right)}^2}} \over {1 + {{\left( {{3 \over 4}} \right)}^2}}} = {7 \over {25}}$
RHS=cos² A -  sin²A= 7/25
So LHS=RHS,so the statement is true

Question 9
In triangle ABC, right-angled at B, if $\tan A = {1 \over {\sqrt 3 }}$
Find the value of:
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C
Solution
$\tan A = {1 \over {\sqrt 3 }}$
${P \over B} = {1 \over {\sqrt 3 }}$
Let $P = k$ and $B = k\sqrt 3 $
Now by Pythagoras theorem
P² + B² =H²
H=2k
(i)
$\sin A \cos C + \cos A \sin C = \left( {{P \over H}} \right)\left( {{B \over H}} \right) + \left( {{B \over H}} \right)\left( {{P \over H}} \right)$
$ = \left( {{{BC} \over {AC}}} \right)\left( {{{BC} \over {AC}}} \right) + \left( {{{AB} \over {AC}}} \right)\left( {{{AB} \over {AC}}} \right)$
$ = {{{k^2}} \over {4{k^2}}} + {{3{k^2}} \over {4{k^2}}} = 1$
(ii)
$\cos A \cos C - \sin A \sin C = \left( {{P \over H}} \right)\left( {{P \over H}} \right) + \left( {{B \over H}} \right)\left( {{B \over H}} \right)$
$ = \left( {{{BC} \over {AC}}} \right)\left( {{{AB} \over {AC}}} \right) - \left( {{{AB} \over {AC}}} \right)\left( {{{BC} \over {AC}}} \right)$
$=0$

Question 10
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.
Solution
Let QR=x andPR=y
Then x+y=25
y=25-x
Now by Pythagorus theorem
x²  + 25= y²
x²  + 25==(25-x)²
Solving it ,we get
X=12 cm
Then y=25-12=13 cm
Now Sin P= 12/13
Cos P=5/13
Tan P=12/5

Question 11
State whether the following are true or false. Justify your answer.
(i) The value of tan A is always less than 1.
(ii) sec A =12/5 for some value of angle A.
(iii) cos A is the abbreviation used for the cosecant of angle A.
(iv) cot A is the product of cot and A.
(v) sin θ =4/3 for some angle
Solution 

1. False. The value of  tan A increase from 0 to ∞.
2. True. The value of sec A increase from 1 to ∞.
3. False .Cos A is the abbreviation used for the cosine of angle A
4. False .cot A is one symbol. We cannot separate it
5. False. The value of sin θ always lies between 0 and 0 and 4/3 > 1

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• Notes
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• NCERT solution
  • trigonometry NCERT Solutions Exercise 8.1
  • trigonometry NCERT Solutions Exercise 8.2
  • trigonometry NCERT Solutions Exercise 8.3

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