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Hong Kong
Stage 4 - Stage 5

Key features of sine and cosine curves (degrees)

Lesson

When looking at the unit circle, that is a circle with the origin at the centre and unit radius, the coordinates of any point on that circle can be described using trigonometry. Specifically a point on the circle at an angle of $\theta$θ anticlockwise from the $x$x-axis has coordinates $\left(\cos\theta,\sin\theta\right)$(cosθ,sinθ).

A point on the unit circle

 

As we move through different values of $\theta$θ the value of $\cos\theta$cosθ and $\sin\theta$sinθ move accordingly between $-1$1 and $1$1. If we plot the values of $\cos\theta$cosθ and $\sin\theta$sinθ according to different values of $\theta$θ on the unit circle, we get the following graphs:

$y=\cos\theta$y=cosθ

 

$y=\sin\theta$y=sinθ

 

Consequently, the graphs of $y=\cos\theta$y=cosθ and $y=\sin\theta$y=sinθ have many properties. Each graph demonstrates repetition. We call the graphs of $y=\cos\theta$y=cosθ and $y=\sin\theta$y=sinθ cyclical and define a cycle as any section of the graph that can be translated to complete the rest of the graph. We also define the period as the length of one cycle. For both graphs, the period is $360^\circ$360°.

An example of a cycle

 

Because of the oscillating behaviour, both graphs have regions where the curve is increasing and decreasing. Remember that we say the graph of a particular curve is increasing if the $y$y-values increase as the $x$x-values increase. Similarly, we say the graph is decreasing if the $y$y-values decrease as the $x$x-values increase.

An example of where $y=\sin x$y=sinx is decreasing

 

In addition, the height of each graph stays between $y=-1$y=1 and $y=1$y=1 for all values of $\theta$θ, since each coordinate of a point on the unit circle can be at most $1$1 unit from the origin.

Practice questions

QUESTION 1

Consider the curve $y=\sin x$y=sinx drawn below and answer the following questions.

Loading Graph...

The curve of the sine function y=sin(x) is plotted on a Cartesian plane. The x-axis ranges from -360 to 360, marked in major intervals of 180 and minor intervals of 45. The y-axis ranges from -2 to 2, marked in major intervals of 1 and minor intervals of 1/2. On the part of the curve visible on the plane, the curve crosses the y-axis at a point. The curve also crosses the x-axis at intervals of 180 on the x values. The crest occur at x=90 and -270. The trough occur at x=-90 and 270. The curve extends indefinitely in both directions along the positive and negative x-axis.
  1. What is the $y$y-intercept? Give your answer as coordinates in the form $\left(a,b\right)$(a,b).

  2. What is the maximum $y$y-value?

  3. What is the minimum $y$y-value?

question 2

Consider the curve $y=\cos x$y=cosx drawn below and determine whether the following statements are true or false.

Loading Graph...

The curve of a cosine function $y=\cos x$y=cosx is plotted. The peaks at $y=1$y=1 occur at $0^\circ$0° at $360^\circ$360° intervals. The troughs at $y=-1$y=1 occur at $180^\circ$180° at $360^\circ$360° intervals. The curve intersects the x-axis at $90^\circ$90°, $270^\circ$270°, $\left(-90\right)^\circ$(90)° and $\left(-270\right)^\circ$(270)°

 

  1. The graph of $y=\cos x$y=cosx is cyclic.

    True

    A

    False

    B
  2. As $x$x approaches infinity, the height of the graph approaches infinity.

    True

    A

    False

    B
  3. The graph of $y=\cos x$y=cosx is increasing between $x=90^\circ$x=90° and $x=180^\circ$x=180°.

    True

    A

    False

    B

question 3

Consider the curve $y=\cos x$y=cosx drawn below and answer the following questions.

Loading Graph...

A cosine curve oscillates above and below the x-axis on a Cartesian coordinate plane. The curve reaches its highest points ($y=1$y=1) at the x-values of  $-360$360, $0$0, and $360$360, but is not explicitly labeled in the image. Its lowest points ($y=-1$y=1) are at the x-values of $-180$180 and $180$180, also not explicitly labeled in the image.
  1. If one cycle of the graph of $y=\cos x$y=cosx starts at $x=\left(-90\right)^\circ$x=(90)°, when does the next cycle start?

  2. In which of the following regions is the graph of $y=\cos x$y=cosx decreasing? Select all that apply.

    $\left(-180\right)^\circ(180)°<x<0°

    A

    $\left(-360\right)^\circ(360)°<x<(180)°

    B

    $0^\circ0°<x<180°

    C

    $180^\circ180°<x<360°

    D
  3. What are the $x$x-values of the $x$x-intercepts in the region $0^\circ0°<x<360°? Give your answers on the same line separated by a comma.

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