### Trigonometric Functions Exercise 01

## Question:

Find the general solution for cos4x=cos2x A x=nπ or nπ/6 B x=nπ or nπ/3 C x=2nπ/3 D x=π

## Answer:

A. x=nπ or nπ/6

## Question:

Find the value of cot(−15π/4)

## Answer:

Step 1: Convert -15π/4 into radians.

Radians = -15π/4 = -7π/2

Step 2: Find the cotangent of -7π/2.

cot(-7π/2) = -cot(7π/2)

Step 3: Use the identity cot(x) = -cot(π - x).

-cot(7π/2) = -cot(π - 7π/2)

Step 4: Simplify.

-cot(7π/2) = -cot(π/2 - 7π/2)

Step 5: Use the identity cot(x) = cot(π - x).

-cot(7π/2) = cot(π/2 - 7π/2)

Step 6: Simplify.

-cot(7π/2) = cot(-5π/2)

Step 7: Use the identity cot(x) = -cot(π + x).

-cot(7π/2) = -cot(π + 5π/2)

Step 8: Simplify.

-cot(7π/2) = -cot(3π/2)

Step 9: Find the cotangent of 3π/2.

cot(-15π/4) = -cot(3π/2) = -√3

## Question:

Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm. (Use π=22/7)

## Answer:

Step 1: Calculate the circumference of the circle using the formula C=2πr C=2π(100) C=200π

Step 2: Calculate the central angle using the formula θ=arc length/radius θ=22/100 θ=0.22

Step 3: Convert the central angle to degree measure using the formula θ (in degree)= θ (in radians) × 180/π θ (in degree)= 0.22 × 180/π θ (in degree)= 12.56°

## Question:

In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

## Answer:

Step 1: Calculate the radius of the circle.

Radius = Diameter/2 Radius = 40 cm/2 Radius = 20 cm

Step 2: Calculate the length of the arc.

Length of arc = (θ/360) x 2πr

Where, θ is the angle subtended by the arc at the center, r is the radius of the circle.

Step 3: Calculate the angle subtended by the arc at the center.

Angle subtended by the arc = (2 x Length of chord)/Radius Angle subtended by the arc = (2 x 20 cm)/20 cm Angle subtended by the arc = 40°

Step 4: Substitute the values in the formula.

Length of arc = (40/360) x 2πr Length of arc = (40/360) x 2π x 20 cm Length of arc = 17.1 cm

## Question:

Find the value of csc(−1410∘)

## Answer:

Step 1: Convert the angle from degrees to radians.

−1410∘ = −1410∘ × (π/180) = −24.7π radians

Step 2: Find the value of csc(−1410∘) using the cosecant function.

csc(−1410∘) = 1/sin(−24.7π)

Step 3: Calculate the value of sin(−24.7π).

sin(−24.7π) = -1

Step 4: Calculate the value of csc(−1410∘).

csc(−1410∘) = 1/(-1) = -1

## Question:

Find the value of sin(−11π/3)

## Answer:

Step 1: Convert -11π/3 into radians.

-11π/3 = -11π/3 * (180/π) = -11*180/3 = -1980/3

Step 2: Use the sin() function to calculate the value of sin(-1980/3).

sin(-1980/3) = -0.8660254037844386

## Question:

If in two circles, arcs of the same length subtend angles 60∘ and 75∘ at the centre, find the ratio of their radii.

## Answer:

Step 1: Draw a diagram of the two circles with arcs of the same length subtending angles of 60° and 75° at the centre.

Step 2: Label the radius of the two circles as r1 and r2.

Step 3: Using the formula for angles subtended by an arc at the centre of a circle, we can write:

60°/2π = r1/2r1

75°/2π = r2/2r2

Step 4: Divide both equations by each other to get:

r1/r2 = (60°/75°) * (2r2/2r1)

Step 5: Simplifying the equation, we get:

r1/r2 = (4/5)

Step 6: Thus, the ratio of the radii of the two circles is 4:5.

## Question:

Find the value of tan 19π/3

## Answer:

Step 1: Find the value of 19π/3 in degrees. (19π/3 = 570°)

Step 2: Use a calculator to find the value of tan 570°. (tan 570° = -1)

## Question:

A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

## Answer:

Step 1: Convert revolutions to radians.

1 revolution = 2π radians

Therefore, 360 revolutions = 360 x 2π = 720π radians

Step 2: Calculate radians per second.

Radians per second = 720π/60 seconds

Step 3: Simplify the equation.

Radians per second = 12π radians

## Question:

Find the angle in radian though which a pendulum swings if its length is 75 cm and the tip describes an arc of length (i) 10 cm (ii) 15 cm (iii) 21 cm

## Answer:

(i) 10 cm

Angle = (10/75) * 2π radians

= (2/15) * π radians

= (2π/15) radians

(ii) 15 cm

Angle = (15/75) * 2π radians

= (3/15) * π radians

= (3π/15) radians

(iii) 21 cm

Angle = (21/75) * 2π radians

= (7/15) * π radians

= (7π/15) radians

## Question:

Find the value of sin 765∘

## Answer:

Step 1: Convert 765° into radians.

1° = π/180

Therefore, 765° = (765 x π)/180 = (4π + 5π)/180 = 9π/180

Step 2: Use the sine formula to calculate the value of sin 765°.

sin 765° = sin (9π/180)

Step 3: Simplify the expression.

sin (9π/180) = sin (π/20)

Step 4: Use a calculator or a table of trigonometric values to find the value of sin (π/20).

sin (π/20) ≈ 0.309017

## Question:

If in two circles, arcs of the same length subtend angles 60o and 75o at the centre, find the ratio of their radii.

## Answer:

Step 1: Draw a diagram of two circles with the angles given.

Step 2: Label the angles as θ1 and θ2.

Step 3: Calculate the ratio of the radii using the formula: Ratio of Radii = tan(θ1)/tan(θ2)

Step 4: Substitute the values of θ1 and θ2 in the formula.

Step 5: The ratio of the radii = tan(60o)/tan(75o)

Step 6: The ratio of the radii = 0.866

## Question:

Find the radian measures corresponding to the following degree measures:(i) 25∘(ii) −47∘30′(iii) 240∘(iv) 520∘

## Answer:

(i) 25∘ = 25 x (π/180) = (25π)/180 radians

(ii) −47∘30′ = -47.5 x (π/180) = (-47.5π)/180 radians

(iii) 240∘ = 240 x (π/180) = (240π)/180 radians

(iv) 520∘ = 520 x (π/180) = (520π)/180 radians

## Question:

Find the degree measures corresponding to the following radian measures (Use π=22/7). (i) 11/16 (ii) −4 (iii) 5π/3 (iv) 7π/6

## Answer:

(i) 11/16 radians = 11/16 x 180/π = 99°

(ii) −4 radians = −4 x 180/π = −226.19°

(iii) 5π/3 radians = 5π/3 x 180/π = 300°

(iv) 7π/6 radians = 7π/6 x 180/π = 315°