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Trigonometric graphs in radians

Sketch sine, cosine and tangent graphs in radians. Find periods, ranges, shifts, exact intercepts and asymptotes with a manual graph explorer and worked practice.

Before you startRadian measure, exact trig values and graph transformations

01 / Read the radian axis

The curves are familiar; the angle labels have changed.

The horizontal input is an angle in radians. A full rotation is 2π, so sine and cosine repeat every 2π. Tangent repeats after π. No degree-to-radian conversion is needed when the axis already uses radians.

Choose a function and move the gold point yourself. The point may disappear at a tangent asymptote or when its value lies outside the shown height; the readout explains which case applies.

Choose a graph and inspect a pointExplore

Period 2π. Range [−1, 1].

At x = π/2: y = 1.

Blue curve: the selected function. Gold point: your chosen input. Dashed red lines mark tangent asymptotes. Values outside the displayed vertical window −3 to 3 are described below the graph.

FunctionInput in steps of π/4 π/2

Watch the period change from 2π to π

Pause, replay or seek freely. The notes explain the same idea and stay in view.

02 / Sketch sine from five key points

One cycle starts at zero and rises.

For y = sin x:
(0,0), (π/2,1), (π,0), (3π/2,−1), (2π,0)

Join these with a smooth curve, then repeat to either side. The domain is all real x, the range is −1 ≤ y ≤ 1, and the smallest positive period is 2π.

The graph is not a zigzag of straight segments. Its peaks and troughs are rounded, with horizontal tangents. For negative inputs, sin(−x) =−sin x, so the graph has rotational symmetry about the origin.

03 / Cosine starts at its maximum

Use the same period with different starting values.

For y = cos x:
(0,1), (π/2,0), (π,−1), (3π/2,0), (2π,1)

The domain is all real x, range [−1,1] and period 2π. Cosine is even: cos(−x) =cos x, so reflection in the y-axis leaves its graph unchanged.

Check the left-hand half-cycleWorked example

cos(−π) =−1; cos(−π/2) =0; cos0 =1

These mirror the values at π, π/2 and 0.

04 / Tangent has separate branches

An asymptote is a missing input, not a vertical piece of the graph.

y = tan x has zeros at x = kπ
and vertical asymptotes at x = π/2 + kπ,
where k is any integer.

One branch passes through (−π/4,−1), (0,0) and (π/4,1), increasing between −π/2 and π/2. Repeat every π. Its range is all real numbers.

Draw each branch separately. Do not join across an asymptote. At x =π/2 the function is undefined, even though values become arbitrarily large in magnitude nearby.

05 / Change amplitude and period

Input and output multipliers do different jobs.

For y = a sin(bx) +d or a cos(bx) +d:
amplitude = |a|; period = 2π/|b|;
range [d −|a|, d +|a|], when a,b ≠0.

For a nonconstant y = a tan(bx) +d, the period isπ/|b|. Tangent has no finite amplitude. For sine or cosine, if a =0 or b =0, deal with the resulting constant function separately; it has no smallest positive period. In a tangent expression, keep its undefined inputs even if its output multiplier is0. Do not erase domain exclusions by multiplying by zero.

Compare three changesWorked example

y =2sin x: amplitude 2, period 2π

Multiply every vertical coordinate by 2.

y =sin2x: amplitude 1, period π

The input completes a full 2π cycle when x increases by π.

y =cos(x/3): amplitude 1, period 6π

The input changes three times more slowly, stretching the graph horizontally.

06 / Shift a trig graph horizontally

Solve the inner expression to locate familiar points.

Sketch y = cos(x −π/3)Worked example

Let u = x −π/3, so x =u +π/3

Each key point moves right by π/3.

Maximum: u =0 gives x =π/3, y =1

A second maximum is at 7π/3.

First zero after that maximum: u =π/2 gives x =5π/6

The next zero is one half-cycle later.

Minimum: u =π gives x =4π/3, y =−1

The period remains2π.

For y =sin(x +π/5), the shift is left by π/5. The sign inside the bracket is opposite to the direction of the point’s horizontal movement.

07 / Combine changes using one mapping

Write the input as b(x −h) to read the horizontal shift.

Sketch y =2sin(2x −π/2) −1Worked example

2x −π/2 =2(x −π/4)

The graph shifts right by π/4, not π/2.

Amplitude2; period π; midline y = −1

Its range is [−3,1].

Inner angle u gives x =u/2 +π/4

Map a whole cycle of sine key points.

(π/4,−1), (π/2,1), (3π/4,−1), (π,−3), (5π/4,−1)

These five points describe one complete cycle.

More generally y = a f(b(x −h)) +d maps (u,v) on y =f(u) to (h +u/b, d +av), for b ≠0. Negative a reflects vertically; negative b reverses the horizontal direction.

08 / Find exact intercepts and asymptotes

Use the inner angle, then return to x.

Intercepts of y =cos(x −π/3) on 0 ≤ x ≤2πWorked example

At x =0: y =cos(−π/3) =1/2

This is the vertical-axis intercept.

For y =0: x −π/3 =π/2 +kπ

Include every integer k whose mapped x lies in the interval.

x =5π/6 or 11π/6

The neighbouring candidates −π/6 and 17π/6 lie outside.

Asymptotes of y =tan(x +π/4)Worked example

x +π/4 =π/2 +kπ

The tangent denominator is zero.

x =π/4 +kπ

Zeros instead satisfy x +π/4 =kπ, so x =−π/4 +kπ.

A vertical translation can change the zero locations, so do not automatically reuse the base function’s intercepts after moving the graph up or down.

09 / State whole families of angles

Use an integer to describe repeated features.

sin x =1 at x =π/2 +2kπ
sin x =−1 at x =3π/2 +2kπ
cos x =1 at x =2kπ
cos x =−1 at x =π +2kπ

In each formula k is an integer, positive, negative or zero. A question restricted to an interval needs a finite list, not just the unrestricted formula.

Where is sin x = −1 on −2π ≤ x ≤3π?Worked example

x =3π/2 +2kπ

Choose integers placing x in the given interval.

What is y =2cos(0x) −1? Does2π/|0| give its period?

Hint

Evaluate cos0 before applying a nonconstant-wave formula.

Worked solution

The function is identically 1. Division by 0 is invalid; every positive shift is a period, so no smallest positive period exists.

11 / Recap

A clear sketch records features, not just a wavy line.

Label the horizontal axis in radians.
Use exact key points for sine and cosine.
Separate tangent branches with asymptotes.
Input multipliers change the period; output multipliers change height.
Factor the input before reading a phase shift.
Find intercepts algebraically and filter by the interval.

Section 1 of 11 · Read the radian axis