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

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

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. 

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