A LEVEL MATHS: Trigonometry

What is the formula for percentage error?

(actual value- approximation)/ actual value

cos90°

0

sin0°

0

tan0°

0

cos0°

1

sin90°

1

tan45°

1

Using addition formula for sin, find sin(A+B), where sinA= 4/5 and sinB= 7/25

1) Look at the addition formula for sin: ≡sinAcosB + sinBcosA 2) Then see that you have been given the ratios for sin: for triangle A: opposite has a value of 4 and the hypoteneus: 5etc. Therefore sonstruct the full triangle and (using pythagoras find the length of the adjacent for both triangles) 3) Through this you can find that cosA= 3/5 and cosB=24/25 4) Subbing all these into the addition formula: answer 117/125

Using addition formulae for tan, show that tan15° = 2- √3

1) Pick out two andles which add up to or substact to make 15°. We must use our known trig values.: tan(60-45) 2) Plug into the addition formulae (tan60 - tan45)/1+ tan60tan45 3) eventually after some cancellng you should get the answer 2-√3

Solve sin3x = -√2/2 or 0≤ x ≤2π

1) multiply the domain by 3: 0≤ 3x ≤6π Then draw a sine graph (normally, not stretched, between 0 and 6π) 2) find sin⁻¹(-√2/2) therefore 3x= -π/4 3) Draw a line onto the sine graph and match up all the points which are found residing within the region 4) 3x= 5π/4, 7π/4, 13π/4, 15π/4, 21π/4, 13π/4 5) x= 5π/12, 7π/12, 13π/12, 15π/4, 21π/12 ....

cos60°

1/2

sin30°

1/2

What is 360° in radians?

2π

What is 120° in radians?

2π/3

What is 270° in radians?

3π/2

tan90°

Asymptote

What is an important rule with regards to dividing trig functions in equations for simplifying?

DON'T DIVIDE TRIG FUNCTIONS unless they are a for functions without zero in the range

Draw the graph of arc tan? What is the domain and what is the range?

Domain: x: -∞<x<∞ Range: -π/2≤ arctanx ≤ π/2

Draw the graph of arcsine? What is the domain and what is the range?

Domain:-1≤x≤1 Range: -π/2≤ arcsinx ≤ π/2

Draw the graph of arc cosine? What is the domain and what is the range?

Domain:-1≤x≤1 Range: 0≤ arcosx≤π

What is another way of writing y= arcosx

If you take cos of both sides: x=cosy cos⁻¹(y) = x

What are two other ways of writing y =arcsinx

If you take sin of both sides: x= siny sin⁻¹(y)=x

What is another way of writing y = arctanx?

If you take tan of both sides: x=tany tan⁻¹(y)= x

Write out 2sinx - 3cosx in the form Rsin(x-a) What are is the maximum value for this?

Maximum value= √13

What is the range of tanx? What is the domain of tanx?

Range: {y: -∞<y<∞} Domain: {x:x∈R} ∩ {x≠(2n-1)π/2, n∈Z}

List the 3 small angle approximations

THIS IS EXPLICITLY FOR RADIANS

Write cos4θ in the form 8cos⁴θ - 8cos²θ +1

Use double angle formulae: cos(2θ +2θ) = 2cos²2θ -1 =2(cos2θ)²-1 =2(2cos²θ -1)² -1 =2(4cos⁴θ - 4cos²θ +1) - 1 =8cos⁴θ - 8cos⁴θ + 1

find an approximation for f(x)= 4cos2xtan3x when x is small

Use small angle approximations: cosθ≈ 1- θ²/2 tanθ≈ θ 4 x (1-(2x)²/2) x (3x) =12x-24x³

Explain the ambiguous case of the

When you have two sides and an angle which is not between the two sides there are two possible triangles which could be created

Write out the R addition formulae

asinθ ± bcosθ = Rsin(θ±α) acosθ ∓ bsinθ= Rcos(θ±α)

Draw a graph of secx Label the asymptotes

asymptotes : x=nπ + π/2 (whenever cos(x)=0)

Draw a graph of cotx label the aymptotes where it cuts the x axis

asymptotes: x=nπ x intercepts: (nπ + π/2)

Draw a graph of cosecx Label the asymptotes coordinates of local minimi/maximi

asymptotes: x=nπ (whenever sinx=0)

What is the cosine rule?

a²=b² + c² - 2bcCosA

i) Approximate cos10° to 6dp and ii) then find the percentage error

i) 1) First you need to change it into radians: 10/180 multiplied by π= π/18 2) Then you need to use the small angle approximation: cosθ≈ 1-θ²/2 Which gives you the answer: 0.984769 (6dp) ii) To find percentage error you need to find the actual value of cos10° which is: 0.984808... Percentage error calculation: (actual value- approximation)/actual value 0.984808-0.984769/0.984808 = 0.004%

What is the quotient trigonometric identity?

sinθ/cosθ

What is range of sinx? What is the domain of sinx?

{y: -1≤y≤1} Domain={x:x∈R}

What is the range of cosx? What is the domain of cosx?

{y: -1≤y≤1} Domain={x:x∈R}

Which are the undefined regions to tanx?

±90, ±270, ±450

What is 180° in radians?

π

What is 90° in radians?

π/2

What is 60° in radians?

π/3

What is 45° in radians?

π/4

What is 30° in radians?

π/6

cos45°

√2/2

sin45°

√2/2

tan60°

√3

cos30°

√3/2

sin60°

√3/2

tan30°

√3/3