Trigonometric Graphs Period = 3600 Amplitude = 1 The horizontal extent of the basic pattern is called the period. Half of the vertical extent is called the amplitude.

Period Period = 3600 Amplitude Amplitude = 1

Period = 1800 Amplitude = undefined The dotted vertical lines are known as asymptotes. The graph approaches but never touches them.

The graph repeats 4 times over 3600. Vertical extent of graph (height) = 6 units. Amplitude = 3.

Sketching Trigonometric Graphs The strategy for sketching trig graphs of the form y = a sin bx  c or y = a cos bx  c is : 1. Start with a simple 2. Put in the scale to match the amplitude a. 3. Slide the graph vertically to match the constant  c.

Step 1. Step 2.

Step 1. Step 2. Step 3.

Step 1. Step 2. Step 3.

Radians Degrees are not the only units used to measure angles. It is often, and usually, useful to measure angles in radians. The angle subtended at the centre of a circle by an arc equal in length to the radius is 1 radian. r r 1 radian r

Hence the radius will fit the circumference 2 times. 1 radian Hence the radius will fit the circumference 2 times. So there are 2 radians in a complete turn.

(cross multiply starting with the ‘x’ term) 1. Convert 1200 into radians. (cross multiply starting with the ‘x’ term) 2 3 This is usually written as

(cross multiply starting with the ‘x’ term) 20 1

Special angles and triangles It is useful and necessary to know exact values of common, and some not so common, trigonometric ratios. If you remember only 2 of these you can work the others out easily. This saves trying to remember them all. Although by the end of the course you will probably have remembered them all as we use them quite a lot in higher maths.

The two ratios that you MUST remember are: (remember TOA) (remember SOH) 2 1 1 1 This gives us exact trig ratios from all 4 quadrants.

The two ratios that you MUST remember are: (remember TOA) (remember SOH) 2 1 1 1 This gives us exact trig ratios from all 4 quadrants.

What is the exact value of (a) sin 3000 (b) cos(-135)0 (C) tan 600 450 -1350

2. Calculate the exact length of the side marked x in the triangle. 600 10 m 5 1

Solving Problems using exact Values An oil tanker is sailing north. At 0115 hours a lighthouse is due east of the tanker. By 0230 hours the lighthouse is 10km from the tanker on a bearing of 1500. Calculate the speed of the tanker. N 1500 T2 300 10 km x Lighthouse T1 Time = 1 hour and 15 minutes

Algebraic solution of equations   Acute value of x.

  A S C T  

A S C T   Acute value of 4x. Remember there are 4 repeats of the sine graph giving 8 solutions over 3600.

  A S C T

 A S C T 

Algebraic Solution of Compound Angle Equations Algebraic solutions of trigonometric equations can be extended to problems involving compound angles such as (3x + 45)0 etc.  A S C T  Acute value of (3x+45).

  A S C T