Introduction to Trigonometry

If cos (40° + x) = sin 30°, find the value of x, provided 40° + x is an acute angle. 

  Prove that    sec² A + co sec²A  = sec²A . cosec A

Write the value of   2 cos²A +2/+ cot²A

 If cot A = 7/8 then what is the value of  (1+ cosA)(1- cosA) ∕ (1- sinA)(1 + sinA

3 If tanA = cot B, prove that A + B = 90°

If sin A = 1/3, then find the value of (2 cot
A + 2).

If sec2 A (1 + sinA)(1-sinA) = k, find the value of k

Evaluate cos 60° sin 30° + sin 60° + cos 30°

If tanA = cot B,What is the Value of A+B ?

What is the Value of tan(90°- A) ?

What is the Value of  Sin² A + Cos²A?

a man is 28.5m away from a tower. His eye level above the ground is
1.5m. The angle of elevation of the tower from his eyes is 45°. Then the
height of a tower is

The angle of elevation of a hovering helicopter as seen from a point 45 m
above a lake is 30c and the angle of depression of its reflection in the lake, as seen
from the same point and at the same time, is 60c. Find the distance of the
helicopter from the surface of the lake.

From the top and foot of a 40 m high tower, the angles of elevation of the top
of a lighthouse are found to be 30cand 60c respectively. Find the height of the
lighthouse. Also find the distance of the top of the lighthouse from the foot of the
tower.

From the top of a tower of height 60 m, the angles of depression of the top
and the bottom of a building are observed to be 30c and 60crespectively. Find the
height of the building

The angles of elevation of an artificial earth satellite is measured from two
earth stations, situated on the same side of the satellite, are found to be 30cand
60c. The two earth stations and the satellite are in the same vertical plane. If the
distance between the earth stations is 4000 km, find the distance between the
satellite and earth. ( 3 = 1.732

A straight highway leads to the foot of a tower. A man standing on the top of
the tower spots a van at an angle of depression of 30c. The van is approaching
the tower with a uniform speed. After 6 minutes, the angle of depression of the
van is found to be 60c. How many more minutes will it take for the van to reach
the tower?

A boy standing on the ground, spots a balloon moving with the wind in a
horizontal line at a constant height . The angle of elevation of the balloon from
the boy at an instant is 60c. After 2 minutes, from the same point of observation,
the angle of elevation reduces to 30c. If the speed of wind is 29

𝟑 m/min. then,
find the height of the balloon from the ground level.

From the top of a lighthouse of height 200 feet, the lighthouse keeper
observes a Yacht and a Barge along the same line of sight. The angles of
depression for the Yacht and the Barge are 45c and 30c respectively. For safety
purposes the two sea vessels should be at least 300 feet apart. If they are less than
300 feet , the keeper has to sound the alarm. Does the keeper have to sound the
alarm

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 30c to 60cas he walks towards the building. Find the
distance he walked towards the building.

A student sitting in a classroom sees a picture on the black board at a height
of 1.5 m from the horizontal level of sight. The angle of elevation of the picture is
30°. As the picture is not clear to him, he moves straight towards the black board
and sees the picture at an angle of elevation of 45°. Find the distance moved by
the student

A person X standing on a horizontal plane, observes a bird flying at a distance
of 100 m from him at an angle of elevation of 30°. Another person Y standing on
the roof of a 20 m high building, observes the bird at the same time at an angle of
elevation of 45°. If X and Y are on the opposite sides of the bird, then find the
distance of the bird from Y.

A person in an helicopter flying at a height of 700 m, observes two objects lying
opposite to each other on either bank of a river. The angles of depression of the
objects are 30° and 45°. Find the width of the river. ( 3 = 1.732 )

A lamp-post stands at the centre of a circular park. Let P and Q be two points
on the boundary such that PQ subtends an angle 90° at the foot of the lamp-post
and the angle of elevation of the top of the lamp post from P is 30°. If PQ = 30 m,
then find the height of the lamp post

Two crows A and B are sitting at a height of 15 m and 10 m in two different
trees vertically opposite to each other . They view a vadai (an eatable) on the
ground at an angle of depression 45° and 60° respectively. They start at the
same time and fly at the same speed along the shortest path to pick up the
vadai. Which bird will succeed in it?

A simple pendulum of length 40 cm subtends 60c at the vertex in one full
oscillation. What will be the shortest distance between the initial position and the
final position of the bob? (between the extreme ends

Suppose two insects A and B can hear each other up to a range of 2 m.. The insect A is on the ground 1 m away from a wall and sees her friend B on the wall,about to be eaten by a spider. If A sounds a warning to B and if the angle of elevation of B from A is 30c, will the spider have a meal or not? (Assume that Bescapes if she hears A alling )

A girl of height 150 cm stands in front of a lamp-post and casts a shadow of
length 150 cm on the ground. Find the angle of elevation of the top of the lamppost

A ramp for unloading a moving truck, has an angle of elevation of 30c. If the
top of the ramp is 0.9 m above the ground level, then find the length of the ramp

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Choose the correct option. Justify your choice.

(i) 9 sec2 A - 9 tan2 A =

(A) 1

(B) 9

(C) 8

(D) 0

(ii) (1 + tan θ + sec θ) (1 + cot θ - cosec θ)

(A) 0

(B) 1

(C) 2

(D) - 1

(iii) (secA + tanA) (1 - sinA) =

(A) secA

(B) sinA

(C) cosecA

(D) cosA

(iv) 

(A) sec2 A

(B) - 1

(C) cot2 A

(D) tan2 A

Evaluate

(i) 

(ii) sin25° cos65° + cos25° sin65°

Write all the other trigonometric ratios of ∠ A in terms of sec A.

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

If A, Band C are interior angles of a triangle ABC then show that

If sec 4A = cosec (A- 20°), where 4A is an acute angle, find the value of A.

If tan A = cot B, prove that A + B = 90°

If tan 2A = cot (A- 18°), where 2A is an acute angle, find the value of A

Show that

(I) tan 48° tan 23° tan 42° tan 67° = 1

(II)cos 38° cos 52° - sin 38° sin 52° = 0

Evaluate

(I) 

(II) 

(III) cos 48° - sin 42°

(IV)cosec 31° - sec 59°

State whether the following are true or false. Justify your answer.

(i) sin(A + B) = sin A + sin B

(ii) The value of sinθincreases as ÃŽÂ¸increases

(iii) The value of cos ÃŽÂ¸increases as ÃŽÂ¸increases

(iv) sinθ= cos ÃŽÂ¸ for all values of ÃŽÂ¸

(v) cot A is not defined for A = 0°

If  and;

0° < A + B ≤90°, A > B find A and B.

Choose the correct option and justify your choice.

(i) 

(A). sin60°

(B). cos60°

(C). tan60°

(D). sin30°

(ii) 

(A). tan90°

(B). 1

(C). sin45°

(D). 0

(iii) sin2A = 2sinA is true when A =

(A). 0°

(B). 30°

(C). 45°

(D). 60°

(iv) 

(A). cos60°

(B). sin60°

(C). tan60°

(D). sin30°

Evaluate the following

(i) sin60° cos30° + sin30° cos 60°

(ii) 2tan245° + cos230° - sin260°

(iii) 

(iv) 

(v)

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 =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 =, for some angle θ

In ΔPQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

In ΔABC, right angled at B. If, find the value of

(i) sin A cos C + cos A sin C

(ii) cos A cos C - sin A sin C

If 3 cot A = 4, Check whether 

If cot θ =, evaluate

(i)  (ii) cot2 θ

If ∠ A and ∠ B are acute angles such that cos A = cos B, then show that

∠ A = ∠ B.

Given sec θ =, calculate all other trigonometric ratios.

Given 15 cot A = 8. Find sin A and sec A

If sin A =, calculate cos A and tan A.

In the given figure find tan P - cot R

In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine

(i) sin A, cos A

(ii) sin C, cos C