## How do we move from triangles to graphs, functions and equations?

### Key questions

1. 1

How are trigonometric functions related to circles?

2. 2

How can we define trigonometric functions for all angles?

3. 3

What are the key features of trigonometric functions and their graphs?

4. 4

How are trigonometric functions related to each other?

5. 5

How can we solve equations and inequalities involving trigonometric functions?

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#### Introducing...

Resource type Title
Building blocks Going round in circles
Food for thought General solutions
Bigger picture From stars to waves
Bigger picture Inverse trigonometric functions

#### Developing...

Resource type Title
Building blocks Muddled trig
Fluency exercise Trig tables
Package of problems Equation or identity? (I)
Many ways problem Can you find... trigonometry edition
Many ways problem Slices of $\pi$
Food for thought Big trig
Food for thought Have a sine
Food for thought How high am I?
Food for thought Inverse or not?
Bigger picture Calculating latitude
Bigger picture Catching waves
Bigger picture Lost but lovely: the haversine
Bigger picture The longitude problem
Resource in action Can you find...trigonometry edition - teacher support
Resource in action Slices of $\pi$ - teacher support
Resource in action Trig tables - teacher support

#### Review questions

Title Ref
Are any of these four trig expressions equal to each other? R7973
As $\theta$ varies, what's the largest this ratio of areas can be? R9770
Can we find the area of a pedal triangle? R8658
Can we find the distance of the tree from the road? R9200
Can we find the lengths and angles in this diagram of a roof? R6144
Can we find the lengths and angles in this pyramid? R6589
Can we prove this trig identity to solve this equation? R5269
Can we show this polynomial has roots $1$, $\cos\theta$ and $\sin\theta$? R8923
Given a picture hanging from a wall, can we find these angles? R5533
How are these three angles connected? R5497
How far are these two ships from the shore? R6049
How many solutions do these simultaneous trig equations have? R7370
How many solutions does $2 \cos^2x + 5\sin x = 4$ have? R9491
How many solutions does $7\sin x +2\cos ^2 x =5$ have? R9837
How many solutions does $\cos (\sin x)=\frac{1}{2}$ have? R8386
How many solutions does $\sin^2x+3\sin x\cos x+2\cos^2 x=0$ have? R6138
How many solutions does this equation involving $\sin^8x$ have? R8136
If $\sin X$ and $\cos Y$ are both negative, what can we say about $X + Y$? R7000
Two swimmers cross a river - can we calculate its width? R7596
What's the distance from a point to a line in the $x\text{-}y$ plane? R9702
What's the result when we reflect $y=\sin x$ twice? R7577
What's this angle of elevation given two others? R7756
When does this equation in $x, y$ and $\theta$ represent a circle? R7378
When is $\sin x$ or $\sin 2x$ larger than a half? R5199
When is $\sin^8 x+\cos^6 x$ equal to $1$? R9817
Which of these area and perimeter statements is true? R8330
Which points in the $x$-$y$ plane satisfy $\sin y=\sin x$? R7374