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E6.4 Trigonometric functions

Syllabus
0580–2028–2029
Topic
E6.4
Level
Extended

Recognise and sketch trigonometric graphs

From 0∘0^\circ to 360∘360^\circ, sine and cosine complete one smooth cycle between −1-1 and 11, while tangent repeats every 180∘180^\circ in separate increasing branches.

Graph Zeros Maxima / minima Other defining features
y=sin⁡xy=\sin x 0∘,180∘,360∘0^\circ,180^\circ,360^\circ max (90∘,1)(90^\circ,1); min (270∘,−1)(270^\circ,-1) period 360∘360^\circ
y=cos⁡xy=\cos x 90∘,270∘90^\circ,270^\circ max (0∘,1)(0^\circ,1) and (360∘,1)(360^\circ,1); min (180∘,−1)(180^\circ,-1) period 360∘360^\circ
y=tan⁡xy=\tan x 0∘,180∘,360∘0^\circ,180^\circ,360^\circ no maximum or minimum vertical asymptotes x=90∘,270∘x=90^\circ,270^\circ; period 180∘180^\circ

Mark the axes and anchor points first. Join sine and cosine anchors with smooth curves, without sharp corners. For tangent, draw dashed vertical asymptotes and four separate branches; each branch rises from negative toward positive values and approaches but never touches or crosses an asymptote.

A graph solves f(x)=kf(x)=k where the curve meets the horizontal line y=ky=k. It also shows sign: sine is positive from 0∘0^\circ to 180∘180^\circ; cosine is positive before 90∘90^\circ and after 270∘270^\circ; tangent is positive in the first and third quadrants.

Do not join tangent across an asymptote or mark x=90∘x=90^\circ and 270∘270^\circ as points on its graph. Sine and cosine stay within −1≤y≤1-1\leq y\leq1; tangent is unbounded. All angles here are degrees, not radians.

Solve trigonometric equations from 0° to 360°

A trigonometric equation can have more than one solution between 0∘0^\circ and 360∘360^\circ. First isolate the trig function, then use its sign and symmetry to find every angle in the interval.

  1. Rearrange to sin⁡x=k\sin x=k, cos⁡x=k\cos x=k or tan⁡x=k\tan x=k. 2. Check the calculator is in degree mode. 3. Find the reference angle β\beta from ∣k∣|k|. 4. Use the function's sign to select the correct quadrants. 5. List every distinct solution in 0∘≤x≤360∘0^\circ\leq x\leq360^\circ and substitute or compare with the graph to check.
Function k>0k>0 k<0k<0
sine β, 180∘−β\beta,\ 180^\circ-\beta 180∘+β, 360∘−β180^\circ+\beta,\ 360^\circ-\beta
cosine β, 360∘−β\beta,\ 360^\circ-\beta 180∘−β, 180∘+β180^\circ-\beta,\ 180^\circ+\beta
tangent β, 180∘+β\beta,\ 180^\circ+\beta 180∘−β, 360∘−β180^\circ-\beta,\ 360^\circ-\beta

Solve 3sin⁡x+1=03\sin x+1=0. Rearranging gives sin⁡x=−13\sin x=-\frac13. The reference angle is β=sin⁡−1(1/3)=19.47…∘\beta=\sin^{-1}(1/3)=19.47\ldots^\circ. Sine is negative in the third and fourth quadrants, so x=180∘+β=199.5∘x=180^\circ+\beta=199.5^\circ or x=360∘−β=340.5∘x=360^\circ-\beta=340.5^\circ to one decimal place.

For sine or cosine, no real solution exists when ∣k∣>1|k|>1. Tangent can equal any real value. Endpoint solutions need care: 0∘0^\circ and 360∘360^\circ are distinct allowed inputs but may give the same trig value, so include each only when it satisfies the equation and interval.