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

Checking and securing understanding of trigonometric ratios

Drawing the sine and cosine graphs

Drawing the tangent graph

The area of any triangle

Calculating the area of any triangle when the height is not known

The sine rule

The cosine rule

Using the sine and cosine rules

Considering the appropriate trigonometric rule

Problem solving with non right-angled trigonometry

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.

In this unit, pupils explore how to apply knowledge in trigonometry to non right-angled triangles.

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?

Interpreting the trigonometric graphs

Lesson outcome