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23 Cards in this Set
- Front
- Back
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What are the two triangles to calculate exact values for sin, cos and tan at 30o, 45o and 60o? |
An isosceles right angled triangle has angles 45o, 45o, 90o and sides 1, 1, √2 An equilateral triangle split in half has angles 30o, 60o, 90o and sides 1, 2, √3. |
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How do you work out trig ratios for angles that are multiples of 30o, 45o and 60o? |
Draw the angle with 0o on the right, and make a right angled triangle to the horizontal line. This will be the ratio. The ratio may be negative. Use the CAST quadrants: SA TC Where A is all. If the letter is in that quadrant, it's positive |
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What do the sin, cos and tan graphs look like? |
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What happens to trig graphs with a transformation like y = f(x)+a? |
The graph will shift up (positive a) /down (negative a) by a |
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What happens to trig graphs with a transformation like y = f(x+a)? |
The graph will shift left (positive a) / right (negative a) by a |
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What happens to trig graphs with a transformation like y = -f(x)? |
It reflects in the x-axis |
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What happens to trig graphs with a transformation like y = f(-x)? |
It reflects in the y-axis |
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What happens to trig graphs with a transformation like y = kf(x)? |
Stretches parallel to y-axis |
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What happens to trig graphs with a transformation like y = f(kx)? |
Stretches parallel to x-axis to the scale factor 1/k where the y-axis is invariant |
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What happens to trig graphs with a transformation like y = f(ax+b)? |
First transform f(ax) then use g(x+b) to transform this to the final answer. Beware that b will need dividing by a before transforming |
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How do you work out the area of a triangle given two sides and the included angle? |
Area△ABC = .5absinC Where a and b are the sides and C is the angle |
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What's the sine rule? |
a/sinA = b/sinB = c/sinC Where a, b and c are the sides and A, B and C are the opposite angles. Can be used to calculate sides given opposite angles, or angles (using inverse sine) |
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Given a side and its opposite angle and one other side, how do you find the remaining angles? |
There are two solutions. Calculate as normal, then look at where else on the sine wave has the same value (180 minus what the calculator spits out) |
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What is the cosine rule? |
a^2 = b^2 + c^2 -2bcCosA |
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What is a radian? |
The angle of sector of a circle where the length of the circumference is equal to the radius, indicated by 1^c. There are 2π radians in 360o. You can use known sections of circles to accurately compare degrees and radians, e.g. 90o = 2π^c / 2 |
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How can you calculate the arc length and the area of a sector? |
Area = θ/360o * πr^2 = θ/2π^c * πr^2 Arc length = θ/360o * 2πr = θ/2π^c * 2πr |
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What are the two short formula with radians for calculating arc lengths and sector areas? |
l = rθ A = 0.5θr^2 |
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How do you calculate the area of a segment? |
Calculate the area of the sector, then the area of the triangle. Take the triangle area from the sector area for the segment area. |
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How do you use the CAST / quadrant diagram to calculate angles? |
It's based on the sin, cos and tan waves. When working out possible values of something like sinθ = 0.5, use it to see which quadrants answers will be in. Draw out the angles on a quadrant and always make the angles to the horizontal line. In this example, it's sin and it's positive so it will be in the 0-90 and 90-180 quadrants. Work it out on a calculator (answer = 30o) and then work out what other angle must be (e.g. 180o-30o=150o). That's the two solutions. Work out all possible angles in the given range. |
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How do you use solve trig equations with multiples of angles |
For example, if you have 2θ, first double the range. Then work out all the way to the end using 2θ rather than θ. Right at the end (and not before), half all the angles to get solutions for θ. |
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How do you solve trig equations with two factors e.g. sinx(1-2cosx) = 0 ? How do you factorise? |
Each factor must equal 0, so solve each in turn to get all solutions. Can pull out terms as normal, but keep trig ratio with variable (e.g. can take sinx as factor, but not sin and x alone). Remember to make one side 0 to do quadratics. |
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What are the two between the trig ratios? |
cos(^2)x + sin(^2)x = 1 sinx / cosx = tanx Can use this to reduce equations to having one trig ratio, e.g. sinx + cosx = 0 can be put in terms of tanx + 1 = 0 (by dividing by cosx) Can also be used to reduce equations to one ratio |
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What's the relationship between sinx and cosx and their negatives? |
cosx = cos(-x) sinx = -sin(-x) |
