Interpreting the trigonometric graphs | Maths | Quizalize & Oak National Academy

Starter quiz

For a repeating function, the __________ is the distance of one repetition of the entire function.

tangent

circle

period ✓

sine

cosine
Match the equivalent values.

$\tan(20°)$

⇔

0.364 (3 s.f) ✓

$\tan(80°)$

⇔

5.67 (3 s.f) ✓

$\tan(97°)$

⇔

-8.14 (3 s.f) ✓

$\tan(105°)$

⇔

-3.73 (3 s.f) ✓
Considering the cosine graph, what is the interval range showing a period?

$0°\leq x \leq 90°$

$0°\leq x \leq 360°$ ✓

$0°\leq x \leq 180°$

$0°\leq x \leq 720°$
Match the values of $x$ to the values of $y$ given $y=x^2$ .

$x = 5$

⇔

$y = 25$ ✓

$x = 2$

⇔

$y = 4$ ✓

$x = -3$

⇔

$y = 9$ ✓

$x = 8$

⇔

$y = 64$ ✓

$x = -6$

⇔

$y = 36$ ✓
Using the function $y=x^2$ , explain how you can show $x = 3$ and $x = -3$ give the same value of $y$ .

Substitute the value for $x$ into $y=x^2$ and both will give 9. ✓

Using inverse functions we know $\pm\sqrt{y} = x$ gives $x = \pm3$ . ✓

Use Desmos to see when $y = 9$ , $x = \pm3$ . ✓

There is only one value of $x$ which maps to $y$ because only $3^2 =9$ .

There is only one value of $x$ as all $x$ values must be positive.
Using the tangent graph, what is the interval range showing a period?

$0°\leq x \leq 90°$

$0°\leq x \leq 360°$

$0°\leq x \leq 180°$ ✓

$0°\leq x \leq 720°$

Using the sine graph, what is the interval range showing a period?

$0°\leq x \leq 90°$

$0°\leq x \leq 360°$ ✓

$0°\leq x \leq 180°$

$0°\leq x \leq 720°$
Using Desmos, draw a cosine graph with the $x$ axis interval of $0°\leq x \leq 360°$ and solve for $\cos(x°) = 1$ .

0° ✓

90°

180°

270°

360° ✓
Using Desmos, draw a sine graph with the $x$ axis interval of $0°\leq x \leq 360°$ and solve for $\sin(x°) = 1$ .

0°

45°

90° ✓

180°

360°
Using Desmos, draw a sine graph with the $x$ axis interval of $0°\leq x \leq 360°$ and estimate the solutions for $\sin(x) = 0.65$ .

41° ✓

139° ✓

141°

49°

21°
How can you check that the solutions to $y=\cos(x°) = 0.42$ within the interval of $0°\leq x \leq 720°$ are correct?

Substitute the value for $x$ into $y=\cos(x°)$ and they will give 0.42... ✓

Using inverse functions we know $\arctan(0.42)$ will give the only answer.

Use Desmos and draw $y=0.42$ and $y=\cos(x°)$ and find the intersections. ✓

There are an infinite number of solutions.
How many solutions will there be when $\tan(x) = 1.6$ in the interval $0°\leq x \leq 360°$ ?

'2' ✓

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Reading solutions from trigonometric graphs is the same as reading solutions from any other graph
Due to the period, there may be more than one solution
It is possible to predict the number of solutions that exist within a given range

Pupils assume that the graph displayed always shows the number of solutions.

It is important to compare the range of values that the graph is displaying against what is being defined in the question.

Trigonometric functions - Trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle for a given angle.
Sine function - The sine of an angle (sin(θ°)) is the y-coordinate of point P on the triangle formed inside the unit circle.
Cosine function - The cosine of an angle (cos(θ°)) is the x-coordinate of point P on the triangle formed inside the unit circle.
Tangent function - The tangent of an angle (tan(θ°)) is the y-coordinate of point Q on the triangle which extends from the unit circle.
Period (of a function) - For a repeating function, the period is the distance of one repetition of the entire function.