In △ A B C , ∠ B = 90 ° and tan A = 1 , then show that 2 sin A cos A = 1

Given, In △ A B C , ∠ A B C = 90 ° and tan A = 1 Here, tan A = B C A B ⇒ 1 = B C A B ⇒ A B = B C Let A B = B C = k , where k is a positive number. Now, applying Pythagoras theorem, we get A C 2 = A B 2 + B C 2 ⇒ A C 2 = k 2 + k 2 ⇒ A C = 2 k 2 = k 2 Now, sin A = B C A C = k k 2 = 1 2 And, cos A = A B A C = k k 2 = 1 2 So, 2 sin A cos A = 2 × 1 2 × 1 2 = 1 Hence proved

If sin A = 3 4 , calculate cos A and c o t A . ( A is an acute angle)

Given, sin A = 3 4 As we know that sin = o p p o s i t e s i d e H y p o t e n u s e Applying Pythagoras theorem A C 2 = A B 2 + B C 2 ⇒ 4 k 2 = A B 2 + 3 k 2 ⇒ 16 k 2 - 9 k 2 = A B 2 ⇒ 7 k 2 = A B ⇒ 7 k 2 = A B cos A = s i d e a d j a c e n t t o ∠ A h y p o t e n u s e = A B A C = 7 k 4 k = 7 4 co t A = cos A sin A = 7 4 3 4 = 7 4 × 4 3 = 7 3

If sec A = 13 12 , calculate sin A and cot A . ( A is an acute angle. )

Given: sec A = 13 12 sec A = Hypotenus Adjacent side to angle A = 13 12 We know that, in a right-angled triangle, Hypotenuse 2 = opposite side 2 + adjacent side 2 ⇒ A C 2 = B C 2 + A B 2 ⇒ 13 2 = B C 2 + 12 2 ⇒ B C 2 = 169 - 144 ⇒ B C 2 = 25 ⇒ B C = 5 cm Now, in a right-angled triangle sin A = opposite side hypotenuse ⇒ sin A = B C A C = 5 13 Also, cot A = adjacent side opposite side ⇒ cot A = A B B C = 12 5

MEDIUM

Earn 100

A $90 cm$ wide door opens on one side of the room at a maximum angle of $90 °$ . Due to shortage of space, a $40 cm$ by $80 cm$ table is kept behind the door along the wall such that it obstructs its path.

(Note: The figure is not to scale)

At what distance from the hinge should the table be kept such that the door opens for a maximum angle of $60 °$ . Show your work. (Use $\sqrt{2} = 1.41, \sqrt{3} = 1.73$ )

Important Questions on Introduction to Trigonometry

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

\tan 7 θ . \tan 3 θ = 1

then

θ =

_____

°

MEDIUM

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

A boy is standing at some distance from a

30 m

tall building and his eye level from the ground is

1.5 m

. The angle of elevation from his eyes to the top of the building increases from

30 °

60 °

as he walks towards the building. Find the distance he walked towards the building.

MEDIUM

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

A

is an acute angle of a

△ A B C

, right-angled at

B

, then the value of

\sin A + \cos A

is:

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

When the length of the shadow of a tree becomes equal to the height of the tree, then the angle of elevation of Sun becomes _____.

HARD

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

A simple pendulum of length

40 cm

subtend

60 °

at the vertex in one full oscillation. What will be the shortest distance between the initial position and final position of the bob?

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

5 \sin θ = 4 \cos θ

then

\tan θ =

_____.

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

The value of

\cos 30 ° \times \sec 30 °

will be

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

\sin A = \frac{3}{5}

then calculate

\cos A

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

△ A B C

∠ B = 90 °

and

\tan A = 1

, then show that

2 \sin A \cos A = 1

MEDIUM

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

If $\sin A = \frac{3}{4},$ calculate $\cos A$ and $c o t A .$ ( $A$ is an acute angle)

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

10 m

long ladder

A C

leans on the wall such that its lower end

(C)

remains

8 m

away from the base of the wall, then

\sin C =

_____.

EASY

Mathematics>Trigonometry>Introduction to Trigonometry>Trigonometric Ratios

The rod $A C$ of a TV disc antenna is fixed at right angles to the wall $A B$ and a rod $C D$ is supporting the disc as shown in the figure. If $A C = 1.5 m$ long and $C D = 3 m$ , find $\tan θ$ .