CAT Exam  >  CAT Notes  >  Quantitative Aptitude (Quant)  >  Basic Trigonometry Concepts

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT PDF Download

What is Trigonometry?

Trigonometry is the branch of mathematics that deals with the study of relationships between sides and angles of a triangle. It is derived from the Greek word 'Trigonon' and 'metron' where , Tri meaning three and Gon means Angle and Metron means Measure.

  • The ratio of the lengths of two sides of a right-angled triangle is called a trigonometric ratio.
  • The trigonometric ratios of a triangle are also called the trigonometric functions. There are six trigonometric ratios. 
  • Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos, and tan.

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

Consider a right-angled triangle ABC, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse are referred to as the adjacent and opposite sides.

  • AC = Hypotenuse of triangle
  • AB = Side adjacent to angle A
  • BC = Side opposite to angle A

    Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

Trignometric Ratios of Right-Angled Triangle ABC with ∠B = 90°


(i) sin A = Sine of ∠A = (Side opposite to ∠A / hypotenuse) = (BC/AC)
(ii) cos A = Cosine of ∠A = (Side adjacent to ∠A / hypotenuse) = (AB/AC)
(iii) tan A = Tangent of ∠A = (Side opposite to ∠A / Side adjacent to ∠A) = (BC/AB)
(iv) cosec A = Cosecant of ∠A = Hypotenuse / Side opposite to ∠A = 1/sin A
(v) sec A = Secant of ∠A = Hypotenuse / Adjacent side = 1/cos A
(vi) cot A = Cotangent of ∠A = Adjacent side / Opposite side = 1/tan A

  • The trigonometry angles which are commonly used in trigonometry problems are  0°, 30°, 45°, 60° and 90°. 
  • The trigonometric ratios such as sine, cosine and tangent of these angles are easy to memorize.
    Example: In a right-angled triangle:
    ⇒ Sin θ = Perpendicular/Hypotenuse
    or θ = sin-1 (P/H)
    ⇒ Similarly,
    θ = cos-1 (Base/Hypotenuse)
    θ = tan-1 (Perpendicular/Base)

Trigonometric Ratios of Specific Angles

  • The trigonometric ratios of some special angles, i.e. 0°, 30°,45°,60°,90° follow a pattern and are easy to remember. 
  • Identifying and remembering these patterns helps in solving problems involving these angles.
    Table: Trigonometric Ratios of Specific AnglesBasic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT
  • Two angles are said to be complementary if their sum is 90°. Thus ϴ and (90 – ϴ) are complementary angles.
  • Representing complementary angles in terms of these standard angles helps in solving a complex problem involving trigonometric ratios.

Question for Basic Trigonometry Concepts
Try yourself:Value of tan 30°/cot 60° is:
View Solution

Trigonometric Ratios of Complementary Angles

  • sin (90° – A) = cos A
  • cos (90° – A) = sin A
  • tan (90° – A) = cot A
  • cot (90° – A) = tan A
  • sec (90° – A) = cosec A
  • cosec (90° – A) = sec A

Basic Trigonometric Identities

  • sin2θ + cos2θ = 1
  • tan2θ + 1 = sec2θ
  • cot2θ + 1 = cosec2θ 
  • sinθ /cosθ = tanθ
  • cosθ / sinθ = cotθ 
  • sin 2θ = 2 sin θ cos θ
  • cos 2θ = cos²θ – sin²θ
  • tan 2θ = 2 tan θ / (1 – tan²θ)
  • cot 2θ = (cot²θ – 1) / 2 cot θ

Question for Basic Trigonometry Concepts
Try yourself:The value of sin θ and cos (90° – θ)
View Solution

Sum and Difference Identities

For angles u and v, we have the following relationships:

  • sin(u + v) = sin(u)cos(v) + cos(u)sin(v)
  • cos(u + v) = cos(u)cos(v) – sin(u)sin(v)
  • tan(u+v) = (tan(u) + tan(v)) / (1−tan(u) tan(v))
  • sin(u – v) = sin(u)cos(v) – cos(u)sin(v)
  • cos(u – v) = cos(u)cos(v) + sin(u)sin(v)
  • tan(u-v) = (tan(u) − tan(v)) / (1+tan(u) tan(v))

Value change with change of angle by π/2 or π or negative sign

When the angle of θ is changed by π/2 or by π the values of the trigonometric functions change in simple ways.

1. Change by π/2:  

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

2. Change by π:

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

3. Change by Sign: 

sin(−θ) = −sinθ, and cos(−θ)=cosθ.

Two Special Triangles

  • There are two special triangles we need to know 45°-45°-90° and 30°-60°-90° triangles. They are depicted in the figures below.

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

  • The figures show how to find the side lengths of those types of these special triangles.
  • In a 45-45-90 triangle ABC shown above, ratio of side AC:BC:AB = 1:1:1/√2
  • In a 60-30-90 triangle ABC shown above, ratio of sides AC:BC:AB = 1: 1/√3: 2

Application of Trigonometry - Heights and Distances

In this problem area, an object stands upright perpendicularly on the surface and from a distance the object is observed. If the angle of elevation is known along with one of the height or distance to the object, the other can be known.

1. Angle of Elevation

When the observer looks up to the top of a tall object, the angle made by the horizontal line through point of observation and the line from the top of the object to the point of observation is the angle of elevation. 

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

We have here, h/d= tanθ

2. Angle of Depression

When an observer looks down from the top of a tall object, we get angle of depression.

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

Example: From a point A on the bridge over a river, the angles of depression of the banks on the opposite sides of the river are 30º and 45º respectively. If the bridge is at a height of 9m above the river surface, find how wide is the river. 

Width of the river is DC. In triangle △ABC, tan 30º= AB/BC300

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT
Solution:

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT 

Important Solved Examples

Let us look at some examples to understand the concept better:

Example 1: Anil looked up at the top of a lighthouse from his boat and found the angle of elevation to be 30º. After sailing in a straight line 50 m towards the lighthouse, he found that the angle of elevation changed to 45º. Find the height of the lighthouse.
a. 25
b. 25√3
c. 25(√3-1)
d. 25(√3+1)

Ans: Option (d) is correct.

Sol:

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

If we look at the above image, A is the previous position of the boat. The angle of elevation from this point to the top of the lighthouse is 30 degrees. After sailing for 50 m, Anil reaches point D from where the angle of elevation is 45 degrees. C is the top of the lighthouse.

Let BD = x
⇨ Now, we know tan 30 degrees = 1/√3 = BC/A
⇨ Tan 45 degrees = 1
⇨  BC = BD = x
Thus, 1/√3 = BC/AB = BC/(AD+DB) = x/(50 + x)
Thus x (3 -1) = 50 or x= 25(3 +1) m. 

Question for Basic Trigonometry Concepts
Try yourself:From a point P on a level ground, the angle of elevation of the top tower is 30°. If the tower is 100 m high, the distance of point P from the foot of the tower is:
View Solution


Example 2: An aeroplane when flying at the height of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 30º and 60º, respectively. Find the distance between the two planes at that instant.

Sol:

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT
Let C and D be the two aeroplanes, and A be the point of observation. Then,
Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT
Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT
Therefore, the distance between the two planes is 6250 m.

Question for Basic Trigonometry Concepts
Try yourself:The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:
View Solution

Example 3: An airplane flying at 3000 m above the ground passes vertically above another plane at an instant when the angle of elevation of the two planes from the same point on the ground are 60º and 45º respectively. The height of the lower plane from the ground is:
a. 1000√3 m
b. 1000/√3 m
c. 500 m
d. 1500(√3+1)

Ans: Option (b) is correct.

Sol:

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

Let the higher plane fly such that at point A the angle of elevation from point D is 60º.
Let the height of the plane flying at a lower level be CD = y (since tan 45º= 1)
⇨ tan 60º = 3000/y
⇨ √3 = 3000/y
⇨ y =  3000/√3 = 1000√3

Example 4: What is the value of (1-CosA) / (1+CosA) given that tan A = ¾?
a. 1/4
b. 9/16
c. 1/9
d. 1/3
Ans: Option (c) is correct.

Sol: We know that tan A = opposite side / adjacent side = 3/4

Therefore, opposite side = 3, adjacent side = 4 and the hypotenuse = 5 

Hence, cos A = 4/5 Therefore, (1-Cos A) / (1 + Cos A) = (1- (4/5)) / (1 + (4/5)) = 1/9 

Example 5: From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30 degree and 45 degree respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

Sol: In the below figure, A and B represent points on the bank on the opposite sides of the river. P is a point on the bridge at a height of 3 m. Length of the side AB is the width of the river.

Now, AB = AD + DB

In right Δ APD, angle A = 30°

So, tan 30° = PD / AD

Or 1/ √3 = 3 / AD or AD = 3√3 m

Also, In right Δ PBD, angle B = 45°. So, BD = PD = 3 m

Now, AB = BD + AD = 3 + 3√3 m

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

Example 6: 3sinx + 4cosx + r is always greater than or equal to 0. What is the smallest value ‘r’ can take? 
a. 5
b. -5
c. 4
d. 3
Ans: Option (a) is correct.

Sol: 3sinx + 4cosx ≥ -r
5(3/5sinx + 4/5cosx) ≥ -r
3/5 = cosA => sinA =4/5
5(sinx cosA + sinA cosx) ≥ -r
5(sin(x + A)) ≥ -r
5sin (x + A) ≥ -r
-1 ≤ sin (angle) ≤ 1
5sin (x + A) ≥ -5
rmin = 5  

Example 7:  Sin2014x + Cos2014x = 1, x in the range of [-5π, 5π], how many values can x take? 
a. 0
b. 10
c. 21
d. 11
Ans: Option (c) is correct.

Sol: We know that Sin2x + Cos2x = 1 for all values of x.
If Sin x or Cos x is equal to –1 or 1, then Sin2014x + Cos2014x will be equal to 1.
Sin x is equal to –1 or 1 when x = –4.5π or –3.5π or –2.5π or –1.5π or –0.5π or 0.5π or 1.5π or 2.5π or 3.5π or 4.5π.
Cosx is equal to –1 or 1 when x = –5π or –4π or –3π or –2π or –π or 0 or π or 2π or 3π or 4π or 5π.
For all other values of x, Sin2014 x will be strictly lesser than Sin2x.
For all other values of x, Cos2014 x will be strictly lesser than Cos2x.
We know that Sin2x + Cos2x is equal to 1. Hence, Sin2014x + Cos2014x will never be equal to 1 for all other values of x. 
Thus there are 21 values.

Example 8: Consider a regular hexagon ABCDEF. There are towers placed at B and D. The angle of elevation from A to the tower at B is 30 degrees, and to the top of the tower at D is 45 degrees. What is the ratio of the heights of towers at B and D? 
a. 1:√3
b. 1:2√3
c. 1:2
d. 3:4√3
Ans: Option (b) is correct.

Sol: When you look at your reflection through a mirror, the image is at a distance equal to the distance between mirror and you. Now, think about what this has to with trigonometry.

Let the hexagon ABCDEF be of side ‘a’. Line AD = 2a. Let towers at B and D be B’B and D’D respectively.
From the given data we know that ∠B´AB = 30° and ∠D´AB = 45°. Keep in mind that the Towers B’B and D´D are not in the same plane as the hexagon.
Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CATIn Triangle B’AB,
Tan∠B´AB = B´AB = B′B/AB = 1/3
=> B’B = a/√3
In Triangle D´AD, tan ∠D´AD = D′D/AD = 1
=> D’D = 2a
The ratio of heights = 1/√3 or 1/2√3
Answer choice (B)

Hence, the answer is 1:2√3

Example 9: A man standing on top of a tower sees a car coming towards the tower. If it takes 20 minutes for the angle of depression to change from 30° to 60°, what is the time remaining for the car to reach the tower?
a. 20√3 minutes
b. 20 minutes
c. 10 minutes
d. 20√3 minutes

Ans: Option (c) is correct.
Sol:  Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT=> a + b = 3b
=> 2b = a
=> b = a2a2

Now since the car takes 20 minutes to travel a distance
Time taken to travel b =a/2 202 = 10 minutes

Example 10: A student is standing with a banner at the top of a 100 m high college building. From a point on the ground, the angle of elevation of the top of the student is 60° and from the same point, the angle of elevation of the top of the tower is 45°. Find the height of the student.
a. 35m
b. 73.2m
c. 50m
d. 75m

Ans: Option (b) is correct.

Sol:
Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

Let BC be the height of the tower and DC be the height of the student.
In rt. ∆ ABC
AB = BC cot 45°
AB = 100 m                                   (i)
In rt. ∆ ABD
AB = BD cot 60°
AB = (BC + CD) cot 60°
AB = (10 + CD)*1/3              (ii)
Equating (i) and (ii)
(10 + CD) * 1/3 = 100
(10 + CD)= 100√3
CD = 100√3 – 100
= 10(1.732 – 1) = 100 * 0.732 = 73.2 m

Example 11: In a trapezium ABCD, AB is parallel to DC, BC is perpendicular to DC and ∠BAD = 45°. If DC = 5 cm, BC = 4 cm, the area of the trapezium in sq. cm is
a. 22 cm2
b. 29 
cm2
c. 28 cm2
d. 32 cm2
Ans: Option (c) is correct.

Sol: Let us consider a trapeziumBasic Trigonometry Concepts | Quantitative Aptitude (Quant) - CATIn ΔAED, ∠AED = 90°, ∠DAE = 45°, 
therefore ∠EDA = 45°.
Clearly ΔAED is an isosceles right triangle and Quadrilateral BCDE is a rectangle.
Therefore, BC = DE = 4cm.
Since ΔAED is an isosceles triangle, AE = ED = 4cm.
Area of the Trapezium = Area of ΔAED + Area of Rectangle BCDE
Area of the Trapezium = 1212 × AE × ED + DC × BC
Area of the Trapezium = 1212 × 4 × 4 + 5 × 4
Area of the Trapezium = 8 + 20
Area of the Trapezium = 28 cm2

The document Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT is a part of the CAT Course Quantitative Aptitude (Quant).
All you need of CAT at this link: CAT
184 videos|131 docs|110 tests

Top Courses for CAT

FAQs on Basic Trigonometry Concepts - Quantitative Aptitude (Quant) - CAT

1. What is the definition of Trigonometry?
Ans.Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. It involves studying the properties and applications of trigonometric functions such as sine, cosine, and tangent.
2. What are the trigonometric ratios for specific angles like 0°, 30°, 45°, 60°, and 90°?
Ans.The trigonometric ratios for the specific angles are as follows: - For 0°: sin(0°) = 0, cos(0°) = 1, tan(0°) = 0 - For 30°: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 - For 45°: sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1 - For 60°: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3 - For 90°: sin(90°) = 1, cos(90°) = 0, tan(90°) is undefined.
3. What are the basic trigonometric identities?
Ans.The basic trigonometric identities include: 1. sin²θ + cos²θ = 1 2. tanθ = sinθ/cosθ 3. 1 + tan²θ = sec²θ 4. 1 + cot²θ = csc²θ These identities are fundamental in simplifying trigonometric expressions and solving equations.
4. How is trigonometry applied in calculating heights and distances?
Ans.Trigonometry is used to calculate heights and distances by employing the concept of angles of elevation and depression. For example, if we know the distance from a point to the base of an object and the angle of elevation to the top of that object, we can use the tangent function to find the height of the object.
5. Can you provide an example of a common trigonometry problem related to heights and distances?
Ans.A common problem could be: "A person is standing 50 meters away from a tree. If the angle of elevation to the top of the tree is 30°, what is the height of the tree?" Using the formula: height = distance * tan(angle), we find height = 50 * tan(30°) = 50 * (1/√3) ≈ 28.87 meters.
184 videos|131 docs|110 tests
Download as PDF
Explore Courses for CAT exam

Top Courses for CAT

Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

,

Important questions

,

Objective type Questions

,

Sample Paper

,

pdf

,

mock tests for examination

,

ppt

,

past year papers

,

study material

,

Previous Year Questions with Solutions

,

video lectures

,

Summary

,

Viva Questions

,

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

,

shortcuts and tricks

,

Free

,

Semester Notes

,

Extra Questions

,

MCQs

,

Basic Trigonometry Concepts | Quantitative Aptitude (Quant) - CAT

,

Exam

,

practice quizzes

;