Trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle for a given angle.

The sine of an angle (sin(θ°)) is the y-coordinate of point P on the triangle formed inside the unit circle.

The cosine of an angle (cos(θ°)) is the x-coordinate of point P on the triangle formed inside the unit circle.

The tangent of an angle (tan(θ°)) is the y-coordinate of point Q on the triangle which extends from the unit circle.

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

Interpreting the trigonometric graphs

Maths

In the prior unit, pupils explored the trigonometric ratios for right-angled triangles in a variety of situations. In this unit, pupils explore what can be done when we have a triangle that is not right-angled.

Non right-angled trigonometry

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

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

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

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

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

How many solutions will there be when $$\tan(x) = 1.6$$ in the interval $$ 0°\leq x \leq 360° $$?

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

Considering the cosine graph, what is the interval range showing a period?

Match the values of $$x$$ to the values of $$y$$ given $$y=x^2$$.

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

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

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