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TrigonometryTrigonometric functions

Exact trig values

Plotting graphs

Sketching graphs

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Here we will learn about *sine graphs*, including how to recognise the graph of the sine function, sketch the sine curve and label important values, and interpret the sine graph.

There are also *sine graphs *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

The sin graph is a visual representation of the sine function for a given range of angles.

The horizontal axis of a trigonometric graph represents the angle, usually written as \theta , and the y -axis is the sine function of that angle.

- The graph does not start at the origin but it does pass through it.
- The graph is continuous and repeats every 360^o
- The maximum value is 1 and the minimum value is -1 .

We will only look at the graph of the sine function in this lesson for angles in degrees although this can also be represented in radians.

Remember that \sin(\theta) is a relationship between the opposite side and the hypotenuse of a right angle triangle:

Let’s look at 3 triangles where we would use the sine ratio to calculate the size of the angle \theta . For each triangle, the hypotenuse is the same but the length of the opposite side and the associated angle change.

Here we can see that as \sin(\theta)=\frac{opp}{hyp} , as the angle \theta increases, the length of the side opposite to the angle also increases. So for each triangle we have:

- Triangle 1: \sin(\theta)=\frac{2}{10}=0.2
- Triangle 2: \sin(\theta)=\frac{6}{10}=0.6
- Triangle 3: \sin(\theta)=\frac{9}{10}=0.9

So what would happen if the opposite side to the angle is equal to 10 ?

\sin(\theta)=\frac{10}{10}=1So when the opposite side is equal to the hypotenuse, we get \sin(\theta) = 1 .

What about when the opposite side is equal to 0 ?

\sin(\theta)=\frac{0}{10}=0So when the opposite side is equal to 0, \; \sin(\theta) = 0.

If we plotted a graph to show the value of \sin(\theta) for each value of \theta between 0^o and 90^o , we get the following graph of the sine function:

Let us add the values of \sin(\theta) for the three triangles from earlier into the graph to show how they would look:

We can now use the graph to find the angle \theta for triangles 1, 2, and 3 :

This graph shows that when \sin(\theta) = 0.2 , \; \theta = 12^o so we have the triangle

This graph shows that when \sin(\theta) = 0.6 , \; \theta = 37^o so we have the triangle

This graph shows that when \sin(\theta) = 0.9 , \; \theta = 64^o so we have the triangle

This shows us that we can use the graph of the sine function to find missing angles in a triangle. More on this later as we have a large problem to resolve. How can the opposite side of a right angle triangle be the same as the hypotenuse, or equal to 0 ?

Unfortunately there is a limit to the use of trigonometric ratios to find angles between 0 and 90^o . For any larger or smaller angles, we need to look at the unit circle.

The unit circle is a circle of radius 1 with the centre at the origin. We can label the values where the circle intersects the axes because we know that the radius of the unit circle is 1 unit.

We can still construct a triangle within the unit circle with the angle starting from the positive x axis.

Looking at the trigonometric ratios of sine and cosine, we can say that:

- \sin(\theta)=\frac{a}{1} \; \text{so} \; a=\sin(\theta)
- \cos(\theta)=\frac{b}{1} \; \text{so} \; b=\cos(\theta)
- \tan(\theta)=\frac{a}{b} \; \text{so} \; \tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}

This means that the width of the triangle is equal to \cos(\theta) and the height of the triangle is equal to \sin(\theta) . The point on the unit circle is therefore the coordinate ( \cos(\theta) , \; \sin(\theta) ):

The y value of the coordinate on the unit circle gives us the value for \tan(\theta) so:

- At the point (1,0) we can say that \sin(0)=0 as the angle \theta would be 0 .
- At the point (0,1) we can say that \sin(90)=1 as the angle \theta is equal to 90^o .
- At the point (-1,0) we can say that \sin(180)=0 as the angle \theta from the positive x axis is now 180^o
- At the point (0,-1) we can say \sin(270)=-1 as the angle \theta is 270^o .
- Continuing the point around the unit circle back to the coordinate (1,0) , we get \sin(360)=0 as \theta has made a full turn back to the positive x axis.

\sin(\theta) is as **periodic function **as we could continue to turn around the origin from the positive x axis beyond 360^o
. We can also turn anticlockwise which would give us the negative value for the value of \sin(\theta) . The period of the function is 360^o .

From the unit circle, we can see that the angle \theta gives a **positive** value for y when the angle \theta is between 0^o and 180^o .

Between 180^o and 360^o , the y value is negative, and so \sin(\theta) is **negative**.

Visualising this using a coordinate axes, we can show when \sin(\theta) is positive and negative:

We are no longer limited to the range of values for \theta as we can continue to turn any angle, in a positive or negative direction and obtain a value for \sin(\theta) . We can therefore expand the graph drawn previously to represent this new set of values on a graph for angles between -360^o and 360^o .

- The graph does not start at the origin but it does pass through it.
- The graph is continuous for both positive and negative values of \theta and has a period of 360^o (this means that the graph repeats every 360^o ).
- The function of \sin(\theta) can be described as a wave, namely the sine wave, with an amplitude of 1 because the function is limited between the values of -1 and 1 . The maximum value is therefore 1 , and the minimum value is -1 .
- The sine wave can have different amplitudes when we transform the function.
- Step by step guide:
**transformations of trigonometric functions.**

You should be able to interpret the unit circle to determine the value of \theta .

Here are a couple of examples to help you.

Top tip: you can remember the coordinate order as alphabetical; x before y and cos before sin to get the coordinate (\cos(\theta), \; \sin(\theta))

Now that we have generated the graph of \sin(\theta) , we need to be able to interpret it to find values of \theta and \sin(\theta) for any value between the range 360^o \leq \theta \leq 360^o (values for \theta between -360^o and 360^o .

The inverse function to \sin(\theta) is \sin^{-1}(\theta) .

The reciprocal to \sin(\theta) is called the cosecant or \cosec(\theta) where \cosec(\theta)=\frac{1}{\sin(\theta)} (not needed for GCSE).

The graphs of \sin \theta and \cos \theta are very similar. In fact, if you applied a phase shift to either graph, you will get the graph of the other function.

Graphing the functions of y=\sin(\theta) (in red) and y=\cos(\theta) (in blue):

If we moved the graph of \cos \theta by adding 90 degrees to every angle (cos(+90)) we get \cos(\theta) . This horizontal shift can be in either direction. We can also carry out a vertical shift, a horizontal stretch, a vertical stretch, a rotation, and more!

Step by step guide: **transforming trigonometric graphs**.

In order to use sine graphs:

- Draw a straight line from the axis of the known value to the sine curve.
- Draw a straight, perpendicular line at the intersection point to the other axis.
- Read the value where the perpendicular line meets the other axis.

Get your free sin graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free sin graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONUse the graph of \sin(\theta) to estimate the value for \theta when \sin(\theta)=1 for 0^o \leq \theta \leq 90^o

**Draw a straight line from the axis of the known value to the sine curve.**

2**Draw a straight, perpendicular line at the intersection point to the other axis.**

3**Read the value where the perpendicular line meets the other axis.**

When \sin(\theta)=1, \; \theta=90^o

Use the graph of \sin(\theta) to estimate the value for \theta when \sin(\theta)=-0.4 for 90^o \leq \theta \leq 270^o

**Draw a straight line from the axis of the known value to the sine** **curve. **

**Draw a straight, perpendicular line at the intersection point to the other axis.**

**Read the value where the perpendicular line meets the other axis.**

When \sin(\theta)=-0.4, \; \theta=204^o

Use the graph of sin(\theta) to estimate the value for \sin(\theta) when \theta=186^o .

**Draw a straight line from the axis of the known value to the sine curve. **

**Draw a straight, perpendicular line at the intersection point to the other axis.**

**Read the value where the perpendicular line meets the other axis.**

When \theta=186^o, \; \sin(\theta)=-0.1 .

Use the graph of \sin(\theta) to estimate the value for \theta when \sin(\theta)=0 for -180 \leq \theta \leq 0^o

**Draw a straight line from the axis of the known value to the sine curve. **

**Draw a straight, perpendicular line at the intersection point to the other axis.**

**Read the value where the perpendicular line meets the other axis.**

When \sin(\theta)=0, \; \theta=180^o and \theta =0^o

Use the graph of \sin(\theta) to estimate the value for \sin(\theta) when \theta=-307^o .

**Draw a straight line from the axis of the known value to the sine curve. **

**Draw a straight, perpendicular line at the intersection point to the other axis.**

**Read the value where the perpendicular line meets the other axis.**

When \theta=-307^o, \; \sin(\theta)=0.8 .

Use the graph of \sin(\theta) to estimate the value for \theta when \sin(\theta)=-0.7 for -360^o \leq \theta \leq 360^o

**Draw a straight line from the axis of the known value to the sine curve. **

**Draw a straight, perpendicular line at the intersection point to the other axis.**

**Read the value where the perpendicular line meets the other axis.**

When \sin(\theta)=-0.6, \; \theta= 143^o, \; \theta=-37^o, \; \theta=217^o \; \text{and} \; \theta=323^o

**Drawing the horizontal line only to one intersection point**

When you are finding the value of \theta using a trigonometric graph, only one value is calculated when there are more points of intersection.

E.g. Only the value of 65^o is read off the graph.

**Sine and cosine graphs switched**

The sine and cosine graphs are very similar and can easily be confused with one another. A tip to remember is that you “sine up” from 0 for the sine graph so the line is increasing whereas you “cosine down” from 1 so the line is decreasing for the cosine graph.

**Asymptotes are drawn incorrectly for the graph of the tangent function**

The tangent function has an asymptote at 90^o because this value is undefined. As the curve repeats every 180^o , the next asymptote is at 270^o and so on.

**The graphs are sketched using a ruler**

Each trigonometric graph is a curve and therefore the only time you are required to use a ruler is to draw a set of axes. Practice sketching each curve freehand and label important values on each axis.

**Value given out of range**

When finding a value of \theta using a trigonometric graph, you must make sure that the value of \theta is within the range specified in the question.

For example, the range of values for \theta is given as 0^o \leq \theta \leq 360^o and only the value of \theta=240^o is written for the solution, whereas the solution \theta=300^o is also correct.

1. Use the graph of \sin(\theta) to calculate the value of \theta when \sin(\theta)=0.26 for 0^o \leq \theta \leq 180^o

\theta=15^o, \;\theta=165^o

\theta=15^o

\theta=165^o

\theta=195^o

2. Use the graph of \sin(\theta) to calculate the value of \theta when \sin(\theta)=-0.5 for -360^o \leq \theta \leq 90^o

\theta=-30^o

\theta=-30^o, \; \theta=-150^o

\theta=-150^o

\theta=-330^o, \; \theta=-210^o

3. Use the graph of \sin(\theta) to calculate the value of \theta when \sin(\theta)=0 for -180^o \leq \theta \leq 180^o

\theta=-180^o

\theta=0^o

\theta=180^o

\theta=180^o, \; 0^o, \; 180^o

4. Use the graph of \sin(\theta) to calculate the value of \sin(\theta) when \theta=200^o

\sin(200)=0.3

\sin(200)=-0.5

\sin(200)=-1

\sin(200)=-0.3

5. Use the graph of \sin(\theta) to calculate the value of \sin(\theta) when \theta=-225^o

\sin(-225)=-0.7

\sin(-225)=0.7

\sin(-225)=1

\sin(-225)=0.2

6. What value for \theta would return the same value for 270^o ?

\sin(270)=sin(90)

\sin(270)=sin(-90)

\sin(270)=sin(0)

\sin(270)=sin(-270)

1. Below is a sketch of the graph of y=\sin(\theta) for 0 \leq \theta \leq 180^o .

If \sin(x)=0.7 , what other value for \theta would return the same value of \sin(\theta) ?

**(1 mark)**

Show answer

180-x

**(1)**

2. (a) Write an equation in terms of \theta for the size of the angle CAB .

(b) Use the graph of y=\sin(\theta) to estimate the value of \theta in triangle ABC .

**(4 marks)**

Show answer

(a)

\sin(\theta)=\frac{12}{13}

**(1)**

**(1)**

(b)

**(1)**

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3. Write down the solutions to the equation \sin(\theta)=-0.2 for 360 \leq \theta \leq 720^o .

**(2 marks)**

Show answer

[-190 \; \text{to} \;-200^o] and [340 \; \text{to} \; 350^o] highlighted on graph

**(1)**

[550-560^o] and [700-710^o]

**(1)**

4. Below are the graphs of y=\sin(\theta) and y=-\cos(\theta). Estimate the solutions for \sin(\theta) = -\cos(\theta) for -360 \leq \theta \leq 360^o .

**(3 marks)**

Show answer

\theta =-45^o, \; -225^o

**(1)**

**(1)**

You have now learned how to:

- recognise, sketch and interpret graphs of trigonometric functions (with arguments in degrees) y = sin \; x for angles of any size.

- Plotting graphs of nonlinear equations
- Transformations of linear graphs
- Transformations of non-linear graphs
- Transforming trigonometric graphs

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