GCSE Mathematics (9-1)
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    13.2 Trigonometric graphs — Practise with help

    Try each question first. Reveal support in order when you need it.

    Warm-up: remember

    1. Work out sin⁡30∘\sin 30^\circ.
    2. Work out cos⁡60∘\cos 60^\circ.
    3. Find the gradient through (0,0)(0,0) and (2,4)(2,4).
    4. Factorise x2−4x^2-4.
    Answers

    1/21/2; 1/21/2; 22; (x−2)(x+2)(x-2)(x+2).

    Practise with help

    1. Sketch y=sin⁡xy=\sin x from 0∘0^\circ to 360∘360^\circ using its quarter-turn values.

    Hint

    Mark the values at 0∘,90∘,180∘,270∘,360∘0^\circ,90^\circ,180^\circ,270^\circ,360^\circ.

    Next step

    The heights in order are 0,1,0,−1,00,1,0,-1,0.

    Full working

    Plot (0,0),(90,1),(180,0),(270,−1),(360,0)(0,0),(90,1),(180,0),(270,-1),(360,0) and join smoothly. The period is 360∘360^\circ.

    2. Solve sin⁡x=0\sin x=0 for 0∘≤x≤360∘0^\circ\leq x\leq360^\circ.

    Hint

    Find where the curve crosses the horizontal axis, including both endpoints.

    Next step

    The sine curve has height zero at the start, halfway point and end of the cycle.

    Full working

    x=0∘,180∘,360∘x=0^\circ,180^\circ,360^\circ.

    3. State the vertical asymptotes of y=tan⁡xy=\tan x from −180∘-180^\circ to 180∘180^\circ.

    Hint

    Tangent is undefined where cosine is zero.

    Next step

    Cosine is zero at −90∘-90^\circ and 90∘90^\circ in this interval.

    Full working

    The asymptotes are x=−90∘x=-90^\circ and x=90∘x=90^\circ. The branches approach these lines without reaching them.

    4. Solve cos⁡x=0\cos x=0 for 0∘≤x≤360∘0^\circ\leq x\leq360^\circ by describing the intersections.

    Hint

    Look where the cosine curve meets the horizontal axis.

    Next step

    The curve crosses at the quarter-turn and three-quarter-turn points.

    Full working

    The intersections occur at x=90∘x=90^\circ and x=270∘x=270^\circ.

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