ASSIGNMENT
TOPIC NAME : Trigonometry
SUBMIT TO : Ms Kiran Rubab
SUBJECT: Teaching of Mathematics
COURSE CODE: EDUC-6124 SEMESTER BS EDUCATION
5th REGULAR(2023-2027)
(Group No. 1) GROUP MEMBERS:
Mehak Noor (EDUC51S23R010)
Abdullah (EDUC51S23R016)
Rimsha (EDUC51S23R040)
INSTITUTE OF EDUCATION
UNIVERSITY OF SARGODHA
Trigonometry
1.Definition of Trigonometry:
Trigonometry is a branch of mathematics that deals with the relationships
between the sides length , and angle of triangles.
2.Measurement of an Angle:
An angle is define as the union of two non-collinera rays with
some common end points.
Vertex : The rays are called arms of the angle and the common
end point is known as vertex of the angle
It is easy if we make an angle by rotating a ray from one
position to another . when we form an angle in this way the
original position of the ray is called the terminal side of the
angle
If the rotation of the ray is anti-clockwise or clockwise , the
angle have +ive or –ive measure respectively
3. Measurement of an angle in sexagesimal system
( degree, mint , sec )
Degree:
A degree, denoted by the symbol "°", is a unit of angular measure. We
divide the circumference of a circle into 360 equal arcs. The angle
subtended at the centre of the circle by one arc is called one degree and is
donated by 1°
The symbol 1°, 1' and 1'' are use to donate a degree , a minute , and a
second respectively
60 sec ( 60') make one minute (1')
60 min ( 60°) make one degree (1°)
90 degree ( 90'' ) make one right angle
360 defree ( 360 '') make 4 right angle
An angle of 360° denotes a complete circle or one revolution. We use
coordinate system to locate any angle to a standard position, where its
initial side is the +ive x-axis and its vertex is the origin
1circle or 1 revolution = 360°
One complete revolution is 360 degrees (360°).
Each degree (°) is divided into 60 minutes (60').
Each minute is further divided into 60 seconds (60'').
4.Conversion from Decimal Degrees (DD) into Degrees, Minutes,
and Seconds (DMS)
Examples:
1. Convert 78.345° to DMS
Degrees: The integer part is 78°.
Minutes: 0.345×60=20.7, so 20'.
Seconds: 0.7×60=42, so 42''.
Final Answer: 78° 20' 42''
2. Convert 32.768° to DMS
Degrees: The integer part is 32°.
Minutes: 0.768×60=46.08, so 46'.
Seconds: 0.08×60=5, so 5''.
Final Answer: 32° 46' 5''
5.Conversion from Degrees, Minutes, and Seconds (DMS) to
Decimal Degrees (DD)
The formula to convert DMS (Degrees, Minutes, Seconds) to Decimal
Degrees (DD) is:
M s
DD= D + 60 + 3600
where:
DD = Degrees
MM = Minutes
SS = Seconds
Example 1: Convert 25° 30' 45'' to decimal form
DD= 25+30/60+45/3600
DD=25+0.5+0.0125
DD=25.5125°
Example 2: Convert 120° 15' 30'' to decimal form
DD=120+15/60+30 /3600
DD=120+0.25+0.0083
DD=120.2583°
6.Conversion between Degrees and Radians:
To convert between degrees and radians, you can use the following
relationships:
Degrees to Radians: Multiply the degree measure by π/180
Radians to Degrees: Multiply the radian measure by 180/π
Example:
Convert 90° to radians:
90°×π /180 = π / 2
Convert π radians to degrees:
π radians×180 / π = 180°
1. Convert an angle from degrees to radians:
The formula for converting degrees to radians is:
Radians=Degrees×π/180
Example: Convert 60° to radians
Radians=60×π/180
= π/3 radians
2. Convert an angle from radians to degrees:
The formula for converting radians to degrees is:
Degrees=Radians×180/ π
Example: Convert 5π/6 radians to degrees
Degrees=5π/6×180/π (π cancel )
=5×180 /6 =150°
5.Circular Measurement of an Angle
The circular measure of an angle is expressed in radians. It is based on the
relationship between the length of an arc and the radius of the circle.
Definition:
The measure of an angle in radians is the ratio of the length of the arc (l) to the
radius (r) of the circle.
OR
The angle made when the radius is wrapped round the circle.
6.Sector of a Circle
A part of the circumference of a circle is called arc
A point of the circle bounded by an arc and a chord is called segment of a
circle
A part of the circle bounded by the two radii and an arc is called sector of
the circle
l
θ= r
where:
θ = Angle in radians
l = Arc length
r = Radius of the circle
Key Facts:
1. One complete revolution (360°) = 2π
2. 180° = π radians.
3. 1 radian ≈ 57.2958°.
Example 1: Finding the Angle in Radians
A pendulum swings along an arc of 20 cm. If the length of the pendulum
(radius) is 50 cm, find the angle it makes in radians.
Solution:
We use the formula:
l
θ= r
where:
s=20 cm (arc length)
r=50 cm (radius)
20
θ= 50
=0.4 radians
Final Answer: 0.4 radians
Example 2: Finding the Arc Length
A wheel of radius 25 cm rotates through an angle of π / 3 radians. Find the arc
length covered by the wheel.
Solution:
We use the formula:
l =r∗θ
where:
r=25cm
π
θ=
3 radians
l=25×π/ 3
s =25π / 3 cm
Final Answer: 25π / 3cm or approximately 26.18 cm
Example 3: Finding the Radius
A car travels along a circular track and covers an arc of 62.8 meters while
making an angle of 2 radians at the center. Find the radius of the track.
Solution:
We use the formula:
r=l / θ
where:
l=62.8m
θ=2 radians
r=62.8 / 2
r =31.4 m
Final Answer: 31.4 meters
6.Relationship Between Degrees and Radians
The relationship between degrees and radians is based on the fact that a full
circle is 360° or 2π radians
180°=π radians
8.Trigonometric Ratios
Trigonometric ratios are the relationships between the sides of a right-angled
triangle and its angles. These ratios are used to define the six basic
trigonometric functions.
Trigonometric Ratio Questions & Answers
Q1: Find the value of sinθ if the opposite side is 6 cm and the hypotenuse is
10 cm.
Solution:
opposite
sinθ= hypotenuse
=6 / 10
=0.6
Answer: 0.6
Q2: If cosθ = 3/5, find secθ.
Solution:
1
secθ= cosθ
1
=3
5
=5 / 3
Answer: 5/3
Q3: In a right-angled triangle, if tanθ = 4/3, find sinθ and cosθ.
Solution:
We know:
opposite
tan θ= adjacent
4
= 3
Using Pythagoras' theorem:
h2=a2+b2
h2¿ √ 4 2+ 32
h2 =√ 16+9
h2¿ √ 25
h= 5
Now,
4 3
sinθ= 5 , cos θ= 5
Answer: sinθ = 4/5, cosθ = 3/5
Q4: If sinθ = 5/13, find cosecθ.
Solution:
1
csc θ= sinθ
1
= 15 ¿
13
¿
=13 / 5
Answer: 13/5
Q5: If cotθ = 7/24, find tanθ.
Solution:
1
tanθ= cotθ
1
7
= 24 ¿
¿
=24 / 17
Answer: 24/7
9.Pythagorean Identities
Formulas:
1. sin2θ +cos2θ = 1
2. 1+tan2θ=sec2θ
3. 1+cot2θ=csc2θ
Questions & Answers:
Step-by-Step Solutions
3
1. If sin θ= 5 , find cos θ using the identity sin2θ+cos2θ=1
Step 1: Use the identity
sin2θ+cos2θ=1
Step 2: Substitute the given value of sin θ\sin\theta
( 35 ) +cos θ=1
2 2
9
= 25 +cos2θ=1
Step 3: Solve for cos2θ
9
cos2θ= 1− 25
25 9
cos2θ= 25 − 25
16
cos2θ = 25
Step 4: Take the square root
√cos2θ = 16
√
4
25
Cos θ = 5
✅ Final Answer: cosθ=45
4
2. If tan θ= 3 , find sec θ
Step 1: Use the identity
1+tan2θ=sec2θ
Step 2: Substitute the given value of tanθ
4
1+ ( 3 ) 2=sec2θ
16
1 + 9 =¿sec2
Step 3: Solve for sec2θ
9 16 25
sec2θ = 9 + 9 = 9
Step 4: Take the square root
√
Sec θ= 25
9
5
√sec2θ = 3
5
✅ Final Answer: sec θ= 3
3. If cot θ=2, find cscθ
Step 1: Use the identity
1+cot2θ=csc2θ
Step 2: Substitute the given value of cotθ
1 +22=csc2θ
1 + 42 = csc2θ
Step 3: Solve for csc2θ
csc2θ=52
Step 4: Take the square root
√csc2θ = √ 5
✅ csc θ=5
Final Answer: csc θ=5
10 Trigonometric Ratios Value Chart
The values of trigonometric functions for standard angles (0°, 30°, 45°, 60°,
90°) are important in solving problems.
Questions & Answers Based on the Table
Q1: Find the value of sin 30° + cos 60°.
Solution:
From the table:
1 1
sin30° = 2 , cos 60°= 2
=sin 30° + cos 60°
1 1
=2 + 2
=1
Answer: 1
Q2: Find the value of tan 45° × cot 30°.
Solution:
From the table:
tan45°=1 , cot 30° = √ 3
= 1× √ 3
= √3
Answer: √3
Q3: Find the value of sec 60° - cosec 30°.
Solution:
From the table:
sec 60°=2 , csc 30°=2
2-2=0
Answer: 0
Q4: Find the value of sin² 45° + cos² 45°.
Solution:
Using identity: sin2θ+cos2θ=1
For θ=45
1 1
=( √2 )2 +( √2 ) 2
=1+ 1
2 2
=1
Answer: 1
Q5: Evaluate tan30∘×tan60∘
Solution:
From the table:
1
tan30° =
√3
, tan60° = √ 3
1
= × √3
√3
=1
Answer: 1
11.Pythagoras Theorem
Pythagoras Theorem Statement:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of
the squares of the other two sides.
h2=a2+b2
Where:
h = Hypotenuse (longest side)
a & b = Other two sides
Q1: The two legs of a right-angled triangle are 6 cm and 8 cm. Find the
hypotenuse.
Solution:
Using Pythagoras theorem:
h2=62+82
h2=36+64
h2=100
h = 10
Answer: 10 cm
Q2: A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm.
Find the other side.
Solution:
Using Pythagoras theorem:
h2=a2+b2
132= 52 + b2
169=25+b2
b2=169−25=144b
b=144
b= 12
Answer: 12 cm
Q3: A ladder of 15 m is placed against a wall. The base of the ladder is 9 m
away from the wall. Find the height at which the ladder touches the wall.
Solution:
Using Pythagoras theorem:
h2=a2+b2
152=92+h2
225=81+h2
h2=225−81
h=144
h = 12
Answer: 12 m
Q4: A triangle has sides of length 7 cm, 24 cm, and 25 cm. Is it a right-
angled triangle?
Solution:
Using Pythagoras theorem:
h2=a2+b2
h^2 = a^2 + b^2
Check if 252=242+72
625=576+49
625 = 576 + 49
625=625(True)
Since the equation holds, the given triangle is a right-angled triangle.
Answer: Yes, it is a right-angled triangle.
Q5: The diagonal of a rectangle is 13 cm, and one of its sides is 5 cm. Find
the other side.
Solution:
A rectangle forms two right-angled triangles with its diagonal as the
hypotenuse.
Using Pythagoras theorem:
h2=a2+b2
132=52+b2
169=25+b2
b2=169−25
b2=144
b= 12
Answer: 12 cm
Multiple Choice Questions (MCQs)
1. What is the value of sin 90°?
a) 0
b) 1
c) ∞
d) -1
✔ Answer: b) 1
2. Which of the following is the reciprocal of cosθ?
a) sinθ
b) tanθ
c) secθ
d) cotθ
✔ Answer: c) secθ
3. If tanθ = 3/4, what is sinθ?
a) 3/5
b) 4/5
c) 5/3
d) 5/4
✔ Answer: a) 3/5
4. What is the value of tan 45°?
a) 0
b) 1
c) √3
d) 1/√3
✔ Answer: b) 1
5. Which of the following is correct?
a) sin²θ + cos²θ = 1
b) sin²θ - cos²θ = 1
c) tan²θ + 1 = cos²θ
d) cot²θ - 1 = sec²θ
✔ Answer: a) sin²θ + cos²θ = 1
6. The value of sin 30° × cos 60° is
a) 1
b) 0.5
c) 0
d) √3
✔ Answer: b) 0.5
7. If secθ = 5/4, what is cosθ?
a) 5/4
b) 4/5
c) 1/5
d) 1/4
✔ Answer: b) 4/5
8. If a ladder of 10 m makes a 60° angle with the ground, what is the
height of the ladder from the ground?
a) 5 m
b) 10√3 m
c) 10 m
d) 10√3/2 m
✔ Answer: d) 10√3/2 m
9. Which of the following is not a trigonometric ratio?
a) sinθ
b) cosθ
c) tanθ
d) cotanθ
✔ Answer: d) cotanθ (should be cotθ)
10.What is the value of sin² 45° + cos² 45°?
a) 0
b) 1
c) 2
d) √2
✔ Answer: b) 1
11.The reciprocal of tanθ is
a) sinθ
b) cosθ
c) cotθ
d) secθ
✔ Answer: c) cotθ
12.If sin A = 3/5, what is the value of cos A?
a) 3/5
b) 4/5
c) 5/3
d) 1/5
✔ Answer: b) 4/5
13.What is the value of cosec 30°?
a) 2
b) 1
c) 0.5
d) 1/2
✔ Answer: a) 2
14.What is the value of tan 60°?
a) 1/√3
b) √3
c) 1
d) 2
✔ Answer: b) √3
15.Which of the following is always true for a right-angled triangle?
a) a² + b² = c²
b) a² - b² = c²
c) a + b = c
d) a - b = c
✔ Answer: a) a² + b² = c²
15 Short Answer Questions on Trigonometry
Q1: Define the six trigonometric ratios.
✔ Answer: The six trigonometric ratios are sine (sinθ), cosine (cosθ), tangent
(tanθ), cosecant (cscθ), secant (secθ), and cotangent (cotθ). They are defined
as follows in a right-angled triangle:
sinθ=opposite / hypotenuse
cosθ=adjacent / hypotenuse
tanθ=opposite adjacent
cscθ=hypotenuse /opposite
secθ=hypotenuse / adjacent
cotθ=adjacent / opposite
Q2: What is the value of sin 45°?
✔ Answer: 1/√2 or √2/2
Q3: Find the value of cos 60°.
✔ Answer: 1/2
Q4: Prove that sin²θ + cos²θ = 1.
✔ Answer: It follows from the Pythagorean Theorem in a right-angled triangle.
Q5: If tanθ = 5/12, find sinθ.
✔ Answer: Using Pythagoras theorem, sinθ = 5/13.
Q7: Convert 30° into radians.
✔ Answer: π/6 radians
Q8: Convert π/4 radians into degrees.
✔ Answer: 45°
Q9: What is the value of sin 0°?
✔ Answer: 0
Q10: If cotθ = 3/4, find secθ.
✔ Answer: 5/4
Q11: What is the value of cosec 90°?
✔ Answer: 1
Q12: What is the value of sec 0°?
✔ Answer: 1
Q13: What is the relationship between sec θ and cos θ?
✔ Answer: sec θ= 1/ cos θ
Q14: What is the relationship between cot θ and tan θ?
✔ Answer: cot θ=1 / tan θ
Q15: Find the height of a tower if the length of its shadow is 10 m and the
angle of elevation is 45°.
✔ Answer: Height = 10 m (since tan 45° = 1, height = shadow length).
