Ratios and Graphs of Trigonometric Functions · Form 5
What do the sine, cosine and tangent graphs show?
A trigonometric graph is a picture of how a ratio changes as the angle grows from 0° to 360°. The sine and cosine graphs roll up and down as a smooth wave between −1 and 1, repeating the same shape every 360°, while the tangent graph climbs steeply and breaks off wherever the cosine is zero.
A graph of a changing ratio
On these graphs the horizontal axis is the angle and the vertical axis is the value of the ratio, so every point simply says 'at this angle, the ratio is this much'. Reading from left to right, you are watching sin, cos or tan change smoothly as the angle turns from 0° to 360°.
The graph is not a new idea, it is the same ratios from the unit circle, laid out in a row so you can see the whole story at once instead of one angle at a time.
The wave that repeats
Because the opposite and adjacent sides can never be longer than the hypotenuse, sine and cosine can never go past 1 or below −1, that is why their waves are trapped in a band of height 1 above and below the axis. They also repeat: turning a full 360° brings you back to the same point on the circle, so the graph copies its shape every 360°, a length called the period.
The cosine graph is the very same wave as sine, just started a quarter-turn (90°) earlier, which is why the two look identical but shifted along.
Where tangent breaks the pattern
Tangent is sine divided by cosine, so wherever cosine hits zero, at 90° and 270°, you would be dividing by zero and the graph simply has no value there. Instead of a gentle wave it races upward, disappears at those angles, and starts again, leaving vertical gaps called asymptotes.
The common misconception is to expect tangent to wave neatly between −1 and 1 like the others; it does not, and its values grow without limit. A quick way to tell the graphs apart: a curve rocking smoothly between −1 and 1 is sine or cosine, while one with repeated steep breaks is tangent.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What are the key features I should label on a sine or cosine graph?
Mark the maximum and minimum values (usually 1 and −1), where the graph crosses the x-axis, and the period, the angle after which the pattern repeats, usually 360°. Exam questions often ask you to sketch the graph across one full period and read off values at specific angles.
Why does the tangent graph look so different from sine and cosine?
Tangent is sine divided by cosine, so wherever cosine equals zero (at 90° and 270°), tangent is undefined and the graph shoots off toward infinity, creating vertical breaks called asymptotes. Between these breaks the curve rises steeply instead of forming a smooth wave.
What mistake do students make when sketching a transformed graph like y = 2 sin x?
They forget the "2" stretches the graph vertically, so the maximum becomes 2 and the minimum becomes −2, while the x-intercepts stay the same. Always check the coefficient in front of sin, cos or tan before sketching, it changes the amplitude, not the period.