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.

Problem solving with non right-angled trigonometry

Maths

Checking and securing understanding of trigonometric ratios

Drawing the sine and cosine graphs

Drawing the tangent graph

Interpreting the trigonometric graphs

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

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

A square has a diagonal length of 8.49 cm. The area of the square, to the nearest integer, is {{}} cm$$^2$$.

A rectangle has two diagonals. The length of each diagonal is 13.4 cm. The diagonals intersect to create two obtuse angles of 127°. Work out the longest length of the rectangle to 2 s.f.

A rectangle has two diagonals. The length of each diagonal is 13.4 cm. The diagonals intersect to create two obtuse angles of 127°. Work out the perimeter to the nearest integer. 

A rectangle has two diagonals. The length of each diagonal is 9.5 cm. The diagonals intersect to create two obtuse angles of 143°. Work out the perimeter to the nearest integer. 

A triangle has three lengths; 11.3 cm, 12.7 cm and 17.0 cm. Identify the three angles. 

A triangle has three lengths; 10.3 cm, 15.8 cm and 8 cm. Identify the three angles when rounded to 3 s.f.

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

$$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$$

Which of these formulae are the sine rule or correct rearrangements of the sine rule?

A triangle has vertices $$A, B$$ and $$C$$, and corresponding lengths $$a, b$$ and $$c$$. $$C$$ = 39.6°, $$c$$ = 6.4 cm and $$a$$ = 9.4 cm. Work out acute angle $$A$$ to 1 significant figure. 

An isosceles triangle has two lengths of 9.22 cm and the third length 14 cm. Work out the angle between the two equal lengths. Give your answer to 1 decimal place. 

An isosceles triangle with lengths 8.6 cm, 8.6 cm and $$x$$ cm has two base angles of 54.5°. Work out the perimeter of the triangle to 1 decimal place. 

The area of a regular hexagon is 93.5 cm$$^2$$. Work out the perimeter of the regular hexagon to the nearest integer. 

Problem solving with non right-angled trigonometry

Lesson outcome