Year 11: Trigonometry

6 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do the sine, cosine and tangent ratios fix the unknown sides and angles of a right-angled triangle, and how do they solve angles of elevation and depression and compass bearings?

Use the trigonometric ratios sine, cosine and tangent to find unknown sides and angles in right-angled triangles, including the exact ratios of 30, 45 and 60 degrees, and apply them to angles of elevation and depression and to compass and true bearings

A focused answer to the Year 11 Maths Advanced dot point on right-angled triangle trigonometry: the three ratios from SOH CAH TOA, choosing the ratio that fits, finding a side and finding an angle, the exact values of 30, 45 and 60 degrees from the two special triangles, angles of elevation and depression, and compass and true bearings, with stage-by-stage diagrams and original worked examples.

17 min answer →

How do you find an unknown side or angle in a triangle that has no right angle, when do you reach for the sine rule and when for the cosine rule, why can the sine rule give two answers (the ambiguous case), and how do you find a triangle's area from two sides and the angle between them?

Establish and apply the sine rule

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

(including the ambiguous case when finding an angle), the cosine rule

a^2 = b^2 + c^2 - 2bc\cos A

to find a side and its rearrangement to find an angle, and the area rule

\text{Area} = \dfrac12 ab\sin C

, to solve problems involving non-right-angled triangles

A focused answer to the Year 11 Maths Advanced dot point on solving non-right triangles: the sine rule for a side or an angle, the ambiguous case where two triangles fit, the cosine rule for a side (SAS) and an angle (SSS), and the area formula $\frac12 ab\sin C$ , with original worked examples in degrees.

19 min answer →

How do you solve a problem set in a three-dimensional solid by finding the one right-angled triangle inside it that carries the answer, and how do you measure the angle a line makes with a plane?

Solve three-dimensional problems involving right-angled triangles, including finding the relevant right-angled triangle inside a solid such as a rectangular prism or pyramid, the angle between a line and a plane, and problems that combine right-triangle results across different planes

A focused answer to the Year 11 Maths Advanced dot point on three-dimensional trigonometry: how to break a solid into right-angled triangles, the angle a space diagonal makes with the base of a box, the angle between a line and a plane, the height of a pyramid apex above the base centre and the angle of a slant edge, with stage-by-stage diagrams and original worked examples.

18 min answer →

How do sine, cosine and tangent make sense for an angle that is not acute, and how do you find the exact value of something like $\cos 150\degree$ or $\tan 210\degree$ using the unit circle, the ASTC sign rule and the related acute angle?

Extend the definitions of sine, cosine and tangent to any angle using the unit circle and the four quadrants, use the ASTC rule for the signs of the ratios, find the related acute angle, determine exact values of the trigonometric functions at angles around the circle, and find one ratio given another together with the quadrant, all in degrees

The Year 11 Maths Advanced dot point on trigonometric functions of any angle: the unit-circle definition beyond acute angles, the four quadrants and the ASTC sign rule, the related acute angle, exact values around the circle, finding one ratio from another given the quadrant, and solving an equation over 0 to 360 degrees, with diagrams and worked examples.

19 min answer →

What do the graphs of $y = \sin x$ , $y = \cos x$ and $y = \tan x$ look like in degrees, what do amplitude and period mean and how do you read them, and how do you solve an equation like $2\cos x = 1$ or $\sin x = -\tfrac12$ for every solution in a stated range such as $0\degree$ to $360\degree$ ?

Sketch the graphs of

y = \sin x

y = \cos x

and

y = \tan x

in degrees over one or more periods, identify their amplitude (where it exists) and period and the key maximum, minimum, zero and intercept points, and solve trigonometric equations of the form

\sin x = k

\cos x = k

and

\tan x = k

for all solutions in a given domain using the graph together with the related acute angle and the ASTC rule

The Year 11 Maths Advanced dot point on trigonometric graphs and equations in degrees: the shape of the sine, cosine and tangent waves over one period, amplitude and period, the key max, min and zero points, and solving equations like 2 cos x = 1 for every solution in 0 to 360 degrees using the related angle and ASTC, with diagrams and worked examples.

20 min answer →

Why is $\sin^2\theta + \cos^2\theta = 1$ true for every angle, how does it link to $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ , and how do you use the two together to prove a trigonometric identity and to simplify a trigonometric expression?

Prove and apply the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

and its rearrangements, and the ratio identity

\tan\theta = \dfrac{\sin\theta}{\cos\theta}

, to prove further trigonometric identities by transforming one side to the other and to simplify trigonometric expressions

A focused answer to the Year 11 Maths Advanced dot point on trigonometric identities: why $\sin^2\theta + \cos^2\theta = 1$ holds for every angle, the rearrangements you will reach for, the ratio identity $\tan\theta = \sin\theta / \cos\theta$ , how to prove an identity by transforming one side, and how to simplify trig expressions, with original worked examples.

17 min answer →