DPST1012 Chap.8 Trigonometric Functions and the Unit Circle
Trigonometric Functions and the Unit Circle
From triangles to functions
Topic 4, in Week 5, turns trigonometry from triangle ratios into functions of any angle.
The syllabus lists radian measure, the angle parameterisation of the unit circle, calculations in right-angled triangles, angles of any magnitude, identities including the reciprocal functions and the Pythagorean identity, sketches using amplitude, period and transformations, and solving equations over a full revolution.
The unit circle and exact values
An angle measured from the positive x-axis meets the unit circle at the point (cos θ, sin θ).
That picture explains the signs in each quadrant, the Pythagorean identity sin²θ + cos²θ = 1, and why sine and cosine repeat every 2π. Radians measure angles by arc length on that circle, with π radians equal to 180 degrees.
A small table of exact values at 0, π/6, π/4, π/3 and π/2, combined with reference angles and quadrant signs, gives the exact values for every standard angle.
Waves and equations
The wave models of Week 2 return with names for each constant. In a sin(b(x − c)) + d, the amplitude is the size of a, the period is 2π divided by b, c shifts the wave sideways and d moves its centre line.
Solving a trigonometric equation means finding every solution in the stated interval: isolate the trigonometric function, find the reference angle, use the quadrant signs, and remember that a double angle such as 2x needs two turns of the circle. The supplied calculator gives only one angle, so it cannot replace this reasoning.
What this chapter covers
- 01
Radian measure
- 02
The unit circle definition of sine and cosine
- 03
Right-angled triangle ratios
- 04
Angles of any magnitude and reference angles
- 05
Reciprocal functions and the Pythagorean identity
- 06
Amplitude, period and wave transformations
- 07
Solving equations over a full revolution
Worked example · free
Find every solution and the wave features
- 1The reference angle where tan = 1 is π/4.
- 1Tangent is negative in the second and fourth quadrants: x = π − π/4 = 3π/4 and x = 2π − π/4 = 7π/4.
- 1For y = 4 cos(3x) − 1, the amplitude is 4 and the period is 2π/3.
- 1The centre line is y = −1, so the range is −1 − 4 to −1 + 4, that is −5 ≤ y ≤ 3.
Key terms
- Radian
- The angle at the centre of a circle that cuts off an arc equal in length to the radius, about 57.3 degrees.
- Unit Circle
- The circle of radius 1 centred at the origin, on which an angle θ marks the point (cos θ, sin θ).
- Reference Angle
- The acute angle between the terminal side of an angle and the x-axis, used to find exact values.
- Amplitude
- Half the distance between the maximum and minimum of a wave, equal to the size of a in a·sin(bx).
- Period
- The horizontal length of one complete cycle of a wave, equal to 2π divided by b in sin(bx).
Trigonometric Functions and the Unit Circle FAQ
Why does calculus need angles in radians?
The simple derivative rules, such as the derivative of sin x being cos x, are only true when x is measured in radians. In degrees an extra conversion factor appears, so the course works in radians from this topic on.
How many solutions should a trigonometric equation have?
It depends on the interval. Over one full revolution, sin x = k has two solutions for most k between −1 and 1. Sketch the graph with the horizontal line y = k and count the crossings before solving.
What is the Pythagorean identity used for?
Finding one ratio from another. If sin θ is known and the quadrant is given, cos θ follows from cos²θ = 1 − sin²θ with the sign chosen by the quadrant, and then tan θ is their quotient.
How do I read the period of a transformed wave?
Divide 2π by the number multiplying x inside the function. A tide that repeats every 12 hours therefore needs that number to be π/6, because 2π divided by π/6 equals 12.
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians, and radians by 180/π to get degrees. So 45 degrees is π/4 and 2π/3 radians is 120 degrees. Keeping common angles as exact multiples of π avoids rounding errors in later steps.
Exam move
Draw the unit circle with the exact values at the standard angles until you can do it in two minutes, then solve one equation per quadrant pair and one double-angle equation each week. Set the supplied calculator model to radians at the start of every practice session.
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