You can give exact sine, cosine and tangent values for the common angles from memory. You can also work out the sign of each ratio from the quadrant alone.
State that cos(pi/3) = 1/2 and sin(pi/3) = root 3 over 2 without a calculator.
Find the reference angle for 210 degrees.
Say which ratios come out negative in the third quadrant.
Worked example
Find sin(210 degrees)
Reference angle: 210 - 180 = 30 degrees
210 degrees is in the third quadrant, where sine is negative
Answer-1/2
Most common mistakeIgnoring the quadrant sign: answering +1/2 because sin(30 degrees) = 1/2, which puts the point above the axis instead of below it.
You can sketch a wave from its formula and pull the amplitude, period, phase shift and midline out of a picture. You can also model repeating things like tides and wheels.
Find the midline of y = 3 sin(x) + 5.
Find the phase shift of y = cos(2x - pi).
Write a wave equation for a Ferris wheel that turns once every 12 minutes.
Worked example
Find the phase shift of y = cos(2x - pi)
Factor the inside: cos(2(x - pi/2))
The shift is whatever is subtracted from x
Answerpi/2 to the right
Most common mistakeReading the shift straight off the -pi: reporting pi to the right, twice the true amount, because the 2 was never factored out first.
You can rewrite one trig expression as another using the Pythagorean, sum, difference and double-angle rules. You can also prove two expressions are equal for every angle.
Replace 1 - sin squared x with cos squared x.
Expand sin(A + B) with the sum rule.
Turn sin(2x) into 2 sin x cos x to make an equation solvable.
Worked example
Simplify (1 - cos squared x) / sin x
1 - cos squared x is sin squared x
sin squared x divided by sin x
Answersin x
Most common mistakeRewriting 1 - cos squared x as (1 - cos x) squared: at x = 60 degrees that gives 0.25 instead of the correct 0.75.
You can solve an equation with sine, cosine or tangent and list every answer inside the interval you were given. You can also add the repeating pattern to describe all of them.
Solve 2 sin x = 1 for x between 0 and 2 pi.
Add 2 pi n to describe every solution, not just the first one.
Factor sin x cos x = sin x rather than dividing, so no roots get lost.
Worked example
Solve 2 sin x = 1 for x from 0 up to 2 pi
sin x = 1/2
Sine is positive in the first and second quadrants
Answerx = pi/6 and x = 5 pi/6
Most common mistakeStopping at the single value the calculator prints: reporting only x = pi/6 and losing the second-quadrant answer 5 pi/6.
You can solve triangles that have no right angle, whether you are given two sides and an angle or all three sides. You can also find a triangle's area without knowing its height.
Use the law of sines when you have a side paired with its opposite angle.
Use the law of cosines when all three sides are known.
Watch for the ambiguous case when two sides and a non-included angle are given.
Worked example
A triangle has sides 7 and 9 with a 40 degree angle between them. Find the third side.
c squared = 49 + 81 - 2(7)(9) cos 40
c squared = 130 - 126(0.766) = 130 - 96.5
Answerc is about 5.79
Most common mistakeUsing the Pythagorean theorem on a triangle with no right angle: computing the square root of 130 = 11.4 instead of 5.79.
Before timing yourself, check whether you can explain Angle measure from a blank page. Then connect it to Unit circle. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Trigonometry calculator can test calculations and representations used in Trigonometry. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Trigonometry practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Angle measure if you are following the full sequence. If that unit feels automatic, open the Trigonometry practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Trigonometry test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Trigonometry calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
