Ratios and Graphs of Trigonometric Functions · Form 5
Ratios and Graphs of Trigonometric Functions: Key Terms
The key Ratios and Graphs of Trigonometric Functions terms you need for SPM Mathematics, defined plainly in English, Malay and Chinese.
- Sine A trigonometric ratio; for an angle, the ratio of the opposite side to the hypotenuse, extended to any angle.
- Trigonometric graph The wave-like graph of sine, cosine or tangent over a range of angles.
- Tangent A trigonometric ratio equal to sine divided by cosine for an angle.
- Cosine A trigonometric ratio; for a right-angled triangle, adjacent over hypotenuse, extended to any angle.
- Angle of elevation The angle measured upward from the horizontal to a line of sight.
- Angle of depression The angle measured downward from the horizontal to a line of sight.
- Special angles The angles 0°, 30°, 45°, 60° and 90° whose trigonometric values are commonly used.
- Quadrant One of the four regions of the coordinate plane, which decides the sign of a trigonometric ratio.
Term pairs students mix up
- Sine graph vs cosine graph, the difference is the starting point: y = sin x begins at 0 (the origin) and first peaks at 90°, while y = cos x begins at its maximum of 1; the two are the same wave shifted 90° apart.
- Angle of elevation vs angle of depression, the difference is direction from the horizontal: elevation is measured upward to an object above eye level, depression is measured downward to an object below it.
- Reference angle vs the actual angle, the difference is size and role: the reference (basic) angle is the acute angle (0°–90°) to the x-axis, while the actual angle (0°–360°) is that reference placed in the correct quadrant.
- Quadrant vs cycle, the difference is span: a quadrant is one 90° region of the axes, while a cycle is one complete 360° repetition of a sine or cosine wave.
- Special angles vs general angles, the difference is exactness: special angles such as 30°, 45° and 60° have exact surd values you memorise, while general angles need a calculator.
How these terms appear in real SPM questions
In real SPM papers the wording tells you the method. 'Solve … for 0° ≤ x ≤ 360°' means list every angle in that interval, not just one.
'Give your answer in surd form' or 'find the exact value' forbids decimals, leave it as 1/√2 or √3/2. 'Sketch the graph of y = … for 0° ≤ x ≤ 360°, hence find the number of solutions of …' wants you to draw a horizontal line and count where it cuts the curve.
'State the coordinates of the maximum point' asks for a point such as (90°, 1). Reading these phrases correctly decides which technique earns the marks.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Is the 'basic angle' the same as the 'reference angle'?
Yes. 'Basic angle' and 'reference angle' are two names for the same thing: the acute angle (between 0° and 90°) measured from the terminal arm to the horizontal x-axis.
SPM Malay papers call it 'sudut rujukan'. Whichever name appears, you find it the same way and use it to place the quadrant sign.
When a question says 'exact value', can I use my calculator's decimal?
No. 'Exact value' means a surd or fraction such as √3/2, not a rounded decimal from the calculator.
The calculator is still useful to check your working, but the written answer must stay exact. Only switch to a decimal when the question explicitly asks for one correct to a stated number of decimal places.
What is the difference between amplitude and period on a trig graph?
Amplitude is the height of the wave from the middle to a peak, for y = sin x it is 1. Period is the horizontal length of one full cycle, for y = sin x and y = cos x it is 360°, and for y = tan x it is 180°.
One measures tallness, the other repeat length.