If A and B are acute angles such that tan A = 1 2 , tan B = 1 3 and tan A + B = tan A + tan B 1 - tan A tan B , show that A + B = 45 °

Given equation is tan A + B = tan A + tan B 1 - tan A tan B , and tan A = 1 2 , tan B = 1 3 Substituting the values in the RHS of the equation, tan A + tan B 1 - tan A tan B = 1 2 + 1 3 ÷ 1 - 1 2 × 1 3 = 3 + 2 6 ÷ 1 - 1 6 = 5 6 ÷ 5 6 = 1 ∴ tan A + B = 1 ⇒ tan A + B = tan 45 ° ⇒ A + B = 45 ° . Hence proved.

If each of α , β and γ is a positive acute angle such that sin α + β - γ = 1 2 , cos β + γ - α = 1 2 and tan γ + α - β = 1 , find the values of α , β and γ .

Given that, sin α + β - γ = 1 2 , cos β + γ - α = 1 2 and tan γ + α - β = 1 From the first equation, sin α + β - γ = 1 2 ⇒ sin α + β - γ = sin 30 ° ⇒ α + β - γ = 30 ° . . . . . ( 1 ) Similarly, cos β + γ - α = 1 2 ⇒ cos β + γ - α = cos 60 ° β + γ - α = 60 ° . . . . . ( 2 ) Again, tan γ + α - β = 1 ⇒ tan γ + α - β = tan 45 ° ⇒ γ + α - β = 45 ° . . . . . ( 3 ) Adding equations ( 1 ) & ( 2 ) , α + β - γ + β + γ - α = 30 ° + 60 ° ⇒ 2 β = 90 ° ⇒ β = 45 ° Now, adding equations 2 & 3 , β + γ - α + γ + α - β = 60 ° + 45 ° ⇒ 2 γ = 105 ° ⇒ γ = 52 1 2 ° And, adding equations 1 & 3 , α + β - γ + γ + α - β = 30 ° + 45 ° ⇒ 2 α = 75 ° ⇒ α = 37 1 2 ° Hence the required values are: α = 37 1 2 ° , β = 45 ° , γ = 52 1 2 ° .

E M B I B E

Modern's abc+ of Mathematics for Class 10>Chapter 8 - Introduction to Trigonometry>EXERCISE>Q 23

MEDIUM

10th CBSE

IMPORTANT

Earn 100

In a right triangle $A B C,$ right triangled at $C$ , if $∠ B = 60 °$ and $A B = 15 cm$ . Find the remaining angles and sides.

Important Questions on Introduction to Trigonometry

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 8 - Introduction to Trigonometry>EXERCISE>Q 24

In a rectangle

A B C D, A B = 20 cm, ∠ B A C = 60 °

, calculate the

B C

and diagonals

A C

and

B D

.

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 8 - Introduction to Trigonometry>EXERCISE>Q 25

If

A

and

B

are acute angles such that

\tan A = \frac{1}{2}, \tan B = \frac{1}{3}

and

\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}

, show that

A + B = 45 °

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 8 - Introduction to Trigonometry>EXERCISE>Q 26

If each of

α, β

and

γ

is a positive acute angle such that

\sin (α + β - γ) = \frac{1}{2}, \cos (β + γ - α) = \frac{1}{2}

and

\tan (γ + α - β) = 1

, find the values of

α, β

and

γ

.

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 8 - Introduction to Trigonometry>EXERCISE>Q 27

A rhombus of side

20 cm

has two angles of

60 °

each. Find the length of the diagonals.

EASY

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 8 - Introduction to Trigonometry>EXERCISE>Q 28

In the following figure $A B C D$ is a rectangle $D C = 8 cm$ and $B C = 6 cm$ . Line segment $C M$ is drawn making an angle $30 °$ with $D C,$ intersecting $A D$ with $M$ . Find the length of $C M$ and $D M$ .