Trigonometry and Calculus Problems

This assignment presents a collection of problems focusing on trigonometry and calculus concepts. It includes questions related to right triangle calculations, conversion between Cartesian and polar coordinates, radians, sinusoidal functions, wave properties, and hyperbolic functions. Each problem provides a solution with clear working steps, making it a valuable resource for students seeking guidance in these mathematical areas.

1) In a right triangle, the measure of one of the angles is 49° and the hypotenuse has a length of
50 cm. Which of the following is the nearest approximation to the length, in cm, of the leg
opposite to this angle?
Ans) 32.8 cm
Working
A= 49°
Cos 49° = AB/AC
0.65 = AB/50
AB = 50*0.65605902899
AB= 32.8 cm
2) A Cartesian coordinate system specifies each point uniquely by a pair of numerical
coordinates, which are the signed distances to the point from two fixed perpendicular directed
lines, measured in the same unit of length.
Ans) True
3) Convert the Cartesian Coordinates (12, 5) in a right angle triangle to Polar Coordinates.
Ans) 13, 22.6^0
Working
Cartesian (x,y) = polar (r, θ)
R = √ x2+ y2
= √ 122+52
= 13
Further, θ = Tan-1 y
x
= Tan-1 12
5
θ = 22.6
4. Express 63 degrees 14 minutes and 51 sec in radian form?

Ans) 1.10 radians
Working
Radians = Degree π/ 180
= (63*3.14)/180
= 1.1
5) What are the three terms used to describe a sinusoidal function?
Ans) Period, Sinusoidal Axis and Amplitude

2) If the tangential speed of an object is 3 meters per second on a circular motion on a path of
radius 2 meters. Find the centripetal acceleration of it?
Ans) 4.5 meter per second
Working
Acceleration = Velocity2/radiun
= 32/2
= 4.5
3). A sound wave travels with a velocity of 330 metres per second and has a frequency of 500
Hz. What is its wavelength?
Ans) 0.66 meters
Working
Wavelength = v/f
= 330/500
= 0.66 meters
4) The phase shift of a Sinusoidal Waveform is the angle θ in degrees or radians is that the
waveform has shifted from a certain reference point along the horizontal zero axis. What is the
equation that represents a negative phase shift where A(m) is the Amplitude of the Wave Form,
wt is the angular frequency of the waveform and phi is the phase angle
Ans) A(t) = A(m) Sin (wt + Phi)

2) What is Cosh^2 (x) + Sinh^2(x) =
Ans) Cosh(2x)
3) Simplify Sin x Cos x
Ans) Sin x