Trigonometric graphs for Edexcel A-level Maths
This page covers the following topics:
1. Unit circle
2. Graphs of trigonometric functions
3. Transformations of trigonometric graphs
4. Graphs of inverse trigonometric functions
5. Graphs of reciprocal trigonometric functions
The unit circle is a circle centred at the origin with a radius of 1. Since it has a radius of 1, it can be used to directly find sine and cosine values, depending on the x and y coordinated of a point. If θ is the angle that the vector to the point makes with the x axis, then the x coordinate is given by cosθ and the y coordinate is given by sinθ.
The graphs for sine and cosine are almost identical, with the only difference being that the cosine graph is the sine graph shifted by 90° to the right. The graphs for sine and cosine both have an amplitude of 1, meaning that the maximum value that the two trigonometric functions can take is 1. They are both 360° periodic, meaning that the graph will be repeated after every 360°.
The cosine graph is symmetric over the y-axis, since when it is flipped across it, the two halves of the graph match up. On the other hand, the graph of sine is symmetric about the origin, since the graph is flipped over both the x-axis and the y-axis. This means that for every x value with a corresponding value of y, −x matches to −y.
Tangent graph is significantly different to both sine and cosine graphs. The graph of tangent is 180° periodic. The graph passes through the origin and tends to infinity both in the positive and the negative direction, and thus its amplitude is undefined. Since tangent is undefined for odd multiples of 90°, at these values of x, the graph of tangent has asymptotes.
If y = f(x) is a trigonometric function, then y = f(ax) is a horizontal stretch of scale factor 1/a, whereas y = af(x) is a vertical stretch with a scale factor a. y = f(−x) is a reflection in the y-axis, whereas y = −f(x) is a reflection in the x-axis. y = f(x) + c is a translation of c units up, whereas y = f(x + c) is a translation of c units to the left.
Both arcsin and arccos have a domain between −1 and 1 inclusive, whereas arctan has an infinite domain. The ranges of arcsin, arccos and arctan are between −π/2 and π/2 inclusive, between 0 and π inclusive and −π/2 and π/2 respectively. The domains and ranges of arcsin and arccos come from the range and domain of sine and cosine respectively. Since tangent has an infinite range, arctan has an infinite domain.
The graphs for the reciprocal trigonometric functions have asymptotes where the standard trigonometric functions have roots. This is because taking the reciprocal of 0 is undefined.
State the maximum amplitude reached by the graph of y = 7sinx + 2.
The graph of y = 7sinx + 2 is a vertical stretch of the y = sinx graph of scale factor 7 and a translation of 2 units up. Since the amplitude of the y = sinx graph is 1, the amplitude of y = 7sinx + 2 is 7 × 1 + 2 = 9.
What is the first positive solution of the equation 2cos(3x) = 0?
The first positive solution of y = cosx is x = 90°. Since y = 2cos(3x) is a vertical stretch of scale factor 2 and a horizontal stretch of scale factor 1/3 of the y = cosx graph, the first solution of 2cos(3x) = 0 is x = 30°.
Explain the magnitude of the minimum value that secant can take.
Secant is given by taking the reciprocal of cosine. The reciprocal gets bigger as the value gets smaller, therefore the minimum reciprocal value is attained at the maximum value. the magnitude of the maximum value of cosine is 1 and the reciprocal of this is 1, therefore the magnitude of the minimum value that secant attains is 1.
Minimum reciprocal is attained at the maximum value.
Explain why the graph of tangent has asymptotes.
Tangent is undefined for odd multiples of 90°, therefore for those x values, the graph will not have a point, thus it has asymptotes there.
Tangent is undefined at odd multiples of 90°.
State the roots of the equation cosx + 3 = 0.
y = cosx + 3 is a translation of the graph of y = cosx 3 units up. This graph does not cross the x-axis, therefore y = cosx + 3 does not have any roots.
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