AP Precalculus: Trigonometric Functions

Setting β = α in the sum identities gives the double-angle identities: sin 2α = 2 sin α cos α and cos 2α = cos² α − sin² α = 1 − 2 sin² α = 2 cos² α − 1. The three forms of cosine let you choose the one that uses only the function you already know. When an equation mixes different angles (such as 2x and x) or different functions (such as sin and cos), use an identity first to rewrite it in terms of a single function and a single angle. Then factor or use the quadratic formula, solve for the trig function, and find the angles in the given interval. Never divide both sides by a trig function, since that can lose solutions; factor it out instead. Finally, check each candidate value: a quadratic in sin x can produce a value larger than 1 in absolute value, which has no angle solution.

1. Periodic Phenomena: Period, Midline, and Amplitude

Trigonometry begins with things that repeat: ocean tides, the seasons, a swinging pendulum, a rider on a Ferris wheel. A function is periodic if its outputs repeat over equal-length intervals of input. The length of the smallest such interval is the period, so f(x + period) = f(x) for every x. Three quantities describe a wave-like periodic function. The midline is the horizontal line halfway between the maximum and minimum: y = (max + min)/2. The amplitude is the distance from the midline to the maximum, which equals (max − min)/2. The period is the horizontal length of one full cycle, for example from one peak to the next. These three numbers are exactly the ones you will later read off a sine or cosine equation, so it pays to be fluent with them from a graph.

2. Angles, Radians, and Arc Length

An angle in standard position has its vertex at the origin and its initial side on the positive x-axis; a counterclockwise rotation is positive and a clockwise rotation is negative. One radian is the measure of a central angle whose arc has the same length as the radius, so for a circle of radius r, the angle θ in radians satisfies θ = s/r, or s = rθ for arc length s. A full turn is 360° = 2π radians, which gives the conversions radians = degrees × π/180 and degrees = radians × 180/π. Angles that end on the same terminal side are coterminal; add or subtract multiples of 360° or 2π to find them. In calculus and in most of AP Precalculus, angles are in radians unless a degree sign is shown, and the calculator must be in radian mode.

3. The Unit Circle and the Trigonometric Functions

Let P(x, y) be any point on the terminal side of an angle θ in standard position, and let r = √(x² + y²) be its distance from the origin, always positive. Then sin θ = y/r, cos θ = x/r, and tan θ = y/x = sin θ/cos θ (undefined when x = 0). On the unit circle, r = 1, so the point is simply (cos θ, sin θ), and x² + y² = 1 becomes cos² θ + sin² θ = 1. The signs follow the quadrant: in Quadrant I all three are positive, in II only sine is positive, in III only tangent is positive, and in IV only cosine is positive. The point does not have to lie on the unit circle; you only need to divide by r.

4. Exact Values of Trigonometric Functions

You should be able to produce the sine, cosine, and tangent of the special angles 0, π/6, π/4, π/3, π/2 and their multiples without a calculator. The coordinates (cos θ, sin θ) are (1, 0), (√3/2, 1/2), (√2/2, √2/2), (1/2, √3/2), (0, 1). For an angle in another quadrant, find its reference angle (the acute angle between the terminal side and the x-axis), use that angle’s values, and then attach the sign for the quadrant. For example, 7π/6 is π/6 past π, so it is in Quadrant III with reference angle π/6. Tangent is sine divided by cosine, and it is undefined at π/2 and 3π/2 where cosine is 0.

5. Graphs of the Sine and Cosine Functions

The graphs of f(θ) = sin θ and g(θ) = cos θ are smooth waves. Both have domain all real numbers, range [−1, 1], amplitude 1, midline y = 0, and period 2π. Sine starts at the origin and rises to its maximum at π/2; it has zeros at every multiple of π and is an odd function, sin(−θ) = −sin θ, so it is symmetric about the origin. Cosine starts at its maximum 1, crosses zero at π/2, and is an even function, cos(−θ) = cos θ, so it is symmetric about the y-axis. Dividing one period into four equal parts of length π/2 gives the five key points that locate a maximum, a minimum, and three midline crossings, and they are the backbone of every sketch. The graph is concave down where the function is above the midline and concave up where it is below, so the points where the curve crosses the midline are inflection points.

6. Sinusoidal Functions: Amplitude, Period, Phase Shift, and Midline

A sinusoidal function is any function of the form y = a sin b(x − c) + d or y = a cos b(x − c) + d. Each constant has a geometric meaning. The amplitude is |a|; if a < 0 the graph is reflected over its midline. The period is 2π/|b|, and the frequency is its reciprocal |b|/(2π). The phase shift c moves the graph c units right (left if c is negative). The vertical shift d gives the midline y = d, so the maximum is d + |a| and the minimum is d − |a|. Be careful when the equation is written as sin(bx − k): factor out b first, because sin(bx − k) = sin b(x − k/b), so the phase shift is k/b and not k. A transformation of the parent graph can be done one step at a time: stretch, reflect, shift horizontally, shift vertically.

7. Graphing by Hand and Writing an Equation from a Graph

Both skills use the same six-question checklist. (1) Amplitude: half the distance from maximum to minimum, a = (max − min)/2. (2) Period: the length of one cycle, which gives b = 2π/period. (3) x-scale: period/4, the spacing of the key points. (4) Midline: d = (max + min)/2. (5) Phase shift: pick a starting point. A sine curve begins at the midline going up; a cosine curve begins at a maximum. The x-value where your chosen curve starts is c. (6) Reflection: if the curve starts the opposite way, make a negative. To graph, plot the five key points of one cycle by moving along the x-scale and then repeat. To write an equation, read the maximum, minimum and one full cycle from the graph, and choose either sine or cosine. Both are correct if the shifts are consistent, and cosine is often easier when the graph has an obvious peak on the y-axis.

8. Sinusoidal Function Context and Data Modeling

Any data that rise and fall in a regular cycle can be modeled with a sinusoid. The method is the six-step checklist applied to a table or scatterplot. Find the maximum and minimum values of the data to get a and d. Decide the period from the context (12 months for a yearly pattern, 24 hours for a daily one) and use b = 2π/period. Then pick a data point at or near the midline on the way up (for sine) or at the maximum (for cosine) to get the phase shift c. Real data are rarely perfect, so the hand model is an approximation; a graphing calculator’s sinusoidal regression (SinReg on a TI-84) finds the best-fitting values, usually in the form a sin(bx + k) + d, which you can factor to compare with your c. Once you have a model, you can substitute a value to make a prediction, or set the model equal to a number and solve to find when something occurs. Always set the calculator to radian mode and comment on whether a prediction is reasonable, since a model is only as good as its fit to the data.

9. The Tangent Function and Its Transformations

Since tan θ = sin θ/cos θ, the tangent graph has vertical asymptotes wherever cos θ = 0, which is at the odd multiples of π/2, that is, θ = π/2 + kπ. Its key features are: period π (one full cycle between consecutive asymptotes), domain all reals except π/2 + kπ, range all real numbers, zeros at every multiple of π, no amplitude and no maximum or minimum, and the function is always increasing on each branch. The graph changes from concave down to concave up at each x-intercept, which is an inflection point. Transformations work as they do for other functions. For y = a tan b(x − c) + d, a is a vertical dilation by |a| (negative a reflects over the x-axis), the period is π/|b|, c is the horizontal shift, and d is the vertical shift. Notice that a does not create an amplitude; it controls how steeply the branches rise. The asymptotes move with the horizontal shift and the period: for the center point (c, d), they sit half a period to each side.

10. Inverse Trigonometric Functions

Sine, cosine, and tangent are not one-to-one, so none has an inverse until the domain is restricted so the graph passes the horizontal line test. The standard restrictions give: y = arcsin x (also written sin⁻¹ x), the angle in [−π/2, π/2] whose sine is x, with domain [−1, 1]; y = arccos x, the angle in [0, π] whose cosine is x, with domain [−1, 1]; and y = arctan x, the angle in (−π/2, π/2) whose tangent is x, with domain all real numbers. The input of an inverse trig function is a ratio and the output is an angle. Be careful: sin⁻¹ x means the inverse function, not 1/sin x. To evaluate one, ask “which angle in the restricted range has this value?” using the unit circle. The restricted range matters: sin⁻¹(sin(7π/6)) is not 7π/6, because 7π/6 is outside [−π/2, π/2]. When a value is not a special ratio, use a calculator in radian mode. Right-triangle sketches help with composites: if θ = arccos x with x > 0, then sin θ = √(1 − x²).

11. Trigonometric Equations and Inequalities

Solve a trigonometric equation in two stages. First, isolate the trigonometric function (for example, sin x = −1/2). Second, find the angles, remembering that the function takes each value in more than one quadrant and that trig functions are periodic. Use the unit circle to get the reference angle (the Quadrant I angle with the same size of value), and place it in every quadrant where the sign is right: sine is negative in III and IV, cosine is negative in II and III, and tangent is negative in II and IV. If the problem gives an interval such as [0, 2π], list only solutions in it. If it gives no interval, add the full period: k·2π for sine and cosine, kπ for tangent, where k is an integer. If the equation is a quadratic in a trig function, factor or use the quadratic formula first, and discard any value outside [−1, 1] for sine and cosine. For inequalities, find the crossing points, then use the graph to see where the curve is above or below the line. If a graphing calculator gives a reference angle, always evaluate it with a positive input so the angle is in Quadrant I, then place it in the correct quadrants yourself.

12. The Secant, Cosecant, and Cotangent Functions

The three remaining trig functions are reciprocals: sec θ = 1/cos θ, csc θ = 1/sin θ, and cot θ = 1/tan θ = cos θ/sin θ. Each is undefined wherever its denominator is zero, and those input values are vertical asymptotes. Secant is undefined where cos θ = 0 (odd multiples of π/2), cosecant where sin θ = 0 (multiples of π), and cotangent where sin θ = 0 (multiples of π). Secant and cosecant have period 2π and range (−∞, −1] ∪ [1, ∞): wherever the cosine or sine graph has a maximum of 1, the reciprocal has a local minimum of 1, and wherever it has a minimum of −1, the reciprocal has a local maximum of −1. Cotangent has period π, no amplitude, and is decreasing between consecutive asymptotes. To evaluate any of these, find the value of the matching sine, cosine, or tangent first, then take the reciprocal.

13. Pythagorean Identities and Proving Identities

An identity is an equation that is true for every input where both sides are defined. Because (cos θ, sin θ) lies on the unit circle, x² + y² = 1 gives the Pythagorean identity sin² θ + cos² θ = 1. Dividing every term by cos² θ gives tan² θ + 1 = sec² θ, and dividing by sin² θ gives 1 + cot² θ = csc² θ. Each can be rearranged, for example sin² θ = 1 − cos² θ. These let you find the other trig values from one: if cos θ is known, then sin θ = ±√(1 − cos² θ) with the sign chosen from the quadrant. To prove an identity, work on one side (usually the more complicated one) until it matches the other, and never move terms across the equals sign. Useful strategies: rewrite everything in sine and cosine, factor, combine fractions over a common denominator, and substitute a known identity. If one side alone does not work, simplify both sides to a common expression.

14. Sum and Difference Identities

The sum identities are sin(α + β) = sin α cos β + cos α sin β and cos(α + β) = cos α cos β − sin α sin β. The difference identities follow by replacing β with −β and using that sine is odd and cosine is even: sin(α − β) = sin α cos β − cos α sin β and cos(α − β) = cos α cos β + sin α sin β. Notice that cosine changes the sign between the two terms while sine keeps it. They have two main uses. First, they give exact values for angles such as π/12 or 7π/12 by writing them as a sum or difference of special angles (for example π/12 = π/3 − π/4). Second, they let you collapse a long expression into a single trig value when you recognize the pattern. Remember that sin(α + β) is not sin α + sin β.

15. Double-Angle Identities and Equations with Identities

Setting β = α in the sum identities gives the double-angle identities: sin 2α = 2 sin α cos α and cos 2α = cos² α − sin² α = 1 − 2 sin² α = 2 cos² α − 1. The three forms of cosine let you choose the one that uses only the function you already know. When an equation mixes different angles (such as 2x and x) or different functions (such as sin and cos), use an identity first to rewrite it in terms of a single function and a single angle. Then factor or use the quadratic formula, solve for the trig function, and find the angles in the given interval. Never divide both sides by a trig function, since that can lose solutions; factor it out instead. Finally, check each candidate value: a quadratic in sin x can produce a value larger than 1 in absolute value, which has no angle solution.

Common Mistakes to Avoid

• Calculating in degree mode. Check that the calculator is in radian mode whenever an angle has π or the problem is in radians.
• Using the period 2π/b for tangent. For tan and cot the period is π/|b|; for sine, cosine, secant, and cosecant it is 2π/|b|.
• Reading the phase shift from sin(bx − k) as k. Factor out b first: the shift is k/b. For sin(2x − π/2) the shift is π/4, not π/2.
• Confusing amplitude with the full height. Amplitude is half the distance from maximum to minimum, and tangent, secant, cosecant, and cotangent do not have an amplitude at all.
• Treating sin⁻¹ x as 1/sin x. The superscript −1 on a function name means inverse (arcsine), and the reciprocal is csc x.
• Giving an angle outside the range of an inverse function, such as writing sin⁻¹(sin(7π/6)) = 7π/6. The output of arcsine must be in [−π/2, π/2], arccosine in [0, π], arctangent in (−π/2, π/2).
• Finding only one solution to an equation like sin x = 1/2. Look in every quadrant where the sign matches, and add the period (k·2π or kπ) when no interval is given.
• Dividing both sides of an equation by a trig function such as sin x, which drops the solutions where it equals 0. Factor it out instead.
• Writing sin(α + β) = sin α + sin β. Trig functions are not linear; use the sum identity.
• Moving terms across the equals sign when proving an identity. Work on one side until it matches the other.

Frequently Asked Questions