Difficulty: Intermediate | Prerequisites: Algebra Foundations notes, Geometry Formulas notes
Trigonometry is woven through nearly every topic in Calculus 1, from limits to derivatives to integrals. If your trig is shaky, you will struggle with derivative rules for sin, cos, and tan, with trig substitution in integrals, and with simplifying expressions. This is the single most-reused prerequisite in the course.
Trigonometric functions relate angles to ratios of sides in right triangles, and extend to the unit circle for all angles. The identities (Pythagorean, double-angle, sum/difference) let you rewrite trig expressions into forms that are easier to differentiate and integrate. Knowing the unit-circle values for standard angles is non-negotiable for calculus.
Sine (sin)
In a right triangle, sin(theta) = opposite / hypotenuse. On the unit circle, sin(theta) is the y-coordinate of the point at angle theta.
Think of it as "how high the point sits" on the unit circle.
Cosine (cos)
In a right triangle, cos(theta) = adjacent / hypotenuse. On the unit circle, cos(theta) is the x-coordinate of the point at angle theta.
Think of it as "how far right the point sits" on the unit circle.
Tangent (tan)
tan(theta) = sin(theta) / cos(theta) = opposite / adjacent. It measures the slope of the terminal side of the angle.
In simple terms, tangent is the ratio of the vertical to the horizontal component.
Cosecant, secant, cotangent (csc, sec, cot)
The reciprocal functions: csc(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), cot(theta) = 1/tan(theta) = cos(theta)/sin(theta).
Think of them as the "flipped" versions of sine, cosine, and tangent respectively.
Unit circle
A circle of radius 1 centred at the origin. Every point on it has coordinates (cos(theta), sin(theta)). It extends the trig functions beyond acute angles to all real numbers.
In simple terms, it is the master reference for all trig values.
Radian
A unit of angle measure where one full revolution = 2·pi radians. One radian is the angle subtended by an arc equal in length to the radius. Calculus always uses radians.
Think of it as measuring angles in terms of the circle's own radius.
Trigonometric identity
An equation involving trig functions that is true for all values of the variable where both sides are defined. Identities are tools for rewriting, not equations to solve.
In simple terms, identities are "always true" relationships you can substitute freely.
sin(theta) = opposite / hypotenuse
cos(theta) = adjacent / hypotenuse
tan(theta) = opposite / adjacent
csc(theta) = hypotenuse / opposite
sec(theta) = hypotenuse / adjacent
cot(theta) = adjacent / opposite
tan(theta) = sin(theta) / cos(theta)
cot(theta) = cos(theta) / sin(theta)
csc(theta) = 1 / sin(theta)
sec(theta) = 1 / cos(theta)
sin(theta) = 1 / csc(theta)
cos(theta) = 1 / sec(theta)
tan(theta) = 1 / cot(theta)
cot(theta) = 1 / tan(theta)
These three are arguably the most important identities in all of calculus.
sin^2(theta) + cos^2(theta) = 1
tan^2(theta) + 1 = sec^2(theta)
cot^2(theta) + 1 = csc^2(theta)
The second and third are obtained by dividing the first by cos^2(theta) and sin^2(theta) respectively.
sin(2theta) = 2·sin(theta)·cos(theta)
cos(2theta) = cos^2(theta) - sin^2(theta) = 2cos^2(theta) - 1 = 1 - 2sin^2(theta)
tan(2theta) = 2·tan(theta) / (1 - tan^2(theta))
The three equivalent forms of cos(2theta) are all useful. Pick whichever eliminates a variable.
sin^2(theta) = (1 - cos(2theta)) / 2
cos^2(theta) = (1 + cos(2theta)) / 2
tan^2(theta) = (1 - cos(2theta)) / (1 + cos(2theta))
These are essential for integrating even powers of sine and cosine.
sin(alpha +/- beta) = sin(alpha)·cos(beta) +/- cos(alpha)·sin(beta)
cos(alpha +/- beta) = cos(alpha)·cos(beta) -/+ sin(alpha)·sin(beta). Note the sign flips for cosine.
tan(alpha +/- beta) = (tan(alpha) +/- tan(beta)) / (1 -/+ tan(alpha)·tan(beta))
sin(alpha) + sin(beta) = 2·sin((alpha+beta)/2)·cos((alpha-beta)/2)
sin(alpha) - sin(beta) = 2·cos((alpha+beta)/2)·sin((alpha-beta)/2)
cos(alpha) + cos(beta) = 2·cos((alpha+beta)/2)·cos((alpha-beta)/2)
cos(alpha) - cos(beta) = -2·sin((alpha+beta)/2)·sin((alpha-beta)/2)
sin(alpha)·sin(beta) = (cos(alpha-beta) - cos(alpha+beta)) / 2
cos(alpha)·cos(beta) = (cos(alpha-beta) + cos(alpha+beta)) / 2
sin(alpha)·cos(beta) = (sin(alpha+beta) + sin(alpha-beta)) / 2
cos(alpha)·sin(beta) = (sin(alpha+beta) - sin(alpha-beta)) / 2
Law of Sines: sin(alpha)/a = sin(beta)/b = sin(gamma)/c
Use when you know two angles and a side, or two sides and an angle opposite one of them.
Law of Cosines: c^2 = a^2 + b^2 - 2ab·cos(gamma) (and the analogous forms for a^2 and b^2)
Use when you know two sides and the included angle, or all three sides.
Angle (degrees) | Angle (radians) | sin | cos | tan |
|---|---|---|---|---|
0 | 0 | 0 | 1 | 0 |
30 | pi/6 | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
45 | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
60 | pi/3 | sqrt(3)/2 | 1/2 | sqrt(3) |
90 | pi/2 | 1 | 0 | undefined |
180 | pi | 0 | -1 | 0 |
270 | 3pi/2 | -1 | 0 | undefined |
360 | 2pi | 0 | 1 | 0 |
All other standard angles (120, 135, 150, 210, 225, 240, 300, 315, 330) are reflections of 30, 45, or 60 into their respective quadrants. Use the reference angle and the sign rules (All Students Take Calculus: all positive in Q1, sin in Q2, tan in Q3, cos in Q4).
Function | Domain | Range | Period |
|---|---|---|---|
y = sin(x) | all reals | [-1, 1] | 2pi |
y = cos(x) | all reals | [-1, 1] | 2pi |
y = tan(x) | all reals, x ≠ pi/2 + k·pi | all reals | pi |
y = csc(x) | all reals, x ≠ n·pi | (-inf, -1] union [1, inf) | 2pi |
y = sec(x) | all reals, x ≠ pi/2 + k·pi | (-inf, -1] union [1, inf) | 2pi |
y = cot(x) | all reals, x ≠ n·pi | all reals | pi |
Trigonometric functions model any phenomenon that repeats: sound waves, alternating current, tides, pendulums, seasonal temperature variation. In engineering, the sum and difference formulas are used to analyse signal interference. The law of cosines is how surveyors and GPS systems calculate distances when direct measurement is impossible.
Students often think sin(2theta) = 2·sin(theta). It does not. sin(2theta) = 2·sin(theta)·cos(theta). The cosine factor is always there.
Students mix up the sign in the cosine sum/difference formula. For cosine, the sign in front of the sin·sin term is the opposite of the sign inside the argument: cos(alpha + beta) uses a minus sign between the terms.
Students forget that tan(theta) is undefined whenever cos(theta) = 0 (at pi/2, 3pi/2, etc.). The same applies to sec(theta).
Students confuse the half-angle formulas with the double-angle formulas. The half-angle formulas are rearrangements of the double-angle formulas, solving for sin^2 or cos^2. They reduce power, not angle.
⚠️ The Pythagorean identity sin^2 + cos^2 = 1 appears in nearly every trig-related problem. You should be able to rearrange it in your sleep.
⚠️ Half-angle (power-reduction) formulas are the standard technique for integrating sin^2(x) and cos^2(x). Expect these on integration exams.
⚠️ Unit-circle values for 0, pi/6, pi/4, pi/3, and pi/2 are tested directly and needed in countless simplification steps. Memorise them.
⚠️ The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These are tested on every single exam.
What is sin(pi/4)? Answer: sqrt(2)/2.
True or false: sin^2(x) + cos^2(x) = 2. Answer: False. It equals 1.
Fill in the blank: cos(2theta) = 2cos^2(theta) - ___. Answer: 1.
What is the period of y = tan(x)? Answer: pi.
True or false: csc(theta) = 1/cos(theta). Answer: False. csc(theta) = 1/sin(theta).
Q: Simplify sin^2(x) + cos^2(x) + tan^2(x).
A: sin^2(x) + cos^2(x) = 1, so the expression becomes 1 + tan^2(x) = sec^2(x).
Q: Use a double-angle formula to find sin(pi/2) given sin(pi/4) = cos(pi/4) = sqrt(2)/2.
A: sin(2·pi/4) = 2·sin(pi/4)·cos(pi/4) = 2·(sqrt(2)/2)·(sqrt(2)/2) = 2·(1/2) = 1. This confirms sin(pi/2) = 1.
Q: Rewrite sin^2(x) in a form suitable for integration.
A: sin^2(x) = (1 - cos(2x)) / 2. This removes the square and makes the antiderivative straightforward.
Q: Find cos(75 degrees) using the sum formula, given 75 = 45 + 30.
A: cos(75) = cos(45)·cos(30) - sin(45)·sin(30) = (sqrt(2)/2)·(sqrt(3)/2) - (sqrt(2)/2)·(1/2) = (sqrt(6) - sqrt(2)) / 4.
Q: In a triangle with sides a = 5, b = 7, and included angle gamma = 60 degrees, find side c.
A: c^2 = 25 + 49 - 2(5)(7)·cos(60) = 74 - 70·(1/2) = 74 - 35 = 39. So c = sqrt(39), approximately 6.24.
Trigonometric identities connect directly to derivatives and integrals. The derivative of sin(x) relies on the limit of sin(h)/h as h approaches 0. Power-reduction formulas are the go-to technique for integrating even powers of trig functions. Trig substitution in integration (covered in Calculus 2) replaces algebraic expressions with trig functions using the Pythagorean identity.
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