Pre-calculus 12
Unit 4 - Trigonometry
Note: Outcome T01 and T02 in the Pre-Calculus 12 Curriculum Document have been moved to Pre-Calculus 11 Curriculum as Outcomes T03 and T04.
Note: Outcome T01 and T02 in the Pre-Calculus 12 Curriculum Document have been moved to Pre-Calculus 11 Curriculum as Outcomes T03 and T04.
Pre-Calculus 12 Pacing Guide - This pacing guide replaces the previous yearly plan. It has been updated to reflect removed outcomes and provide flexibility for responsive instruction.
Pre-calculus 12 Desmos Activity Collection - A collection of online student Desmos activities organized by unit.
EAL Support - Desmos offers a free suite of math software tools, including the Desmos Graphing Calculator and Scientific Calculator, as well as free digital classroom activities. Click on the globe in the tool bar to access the site in other languages.
T03 - Simplifying trigonometric expressions involving rationalizing requires supplementary practice questions.
T04 - Students are not expected to transform y = tan(x) or solve problems that involve the tangent function.
T06 - It is expected that students will be provided with a list of the identities for a formal assessment.
Note that T01 and T02 should have been covered in Pre-calculus 11. If not, these outcomes will need to be included here.
T03 Students will be expected to solve problems, using the six trigonometric ratios for angles expressed in radians and degrees. [ME, PS, R, T, V]
T03.01 Determine, with technology, the approximate value of a trigonometric ratio for any angle with a measure expressed in either degrees or radians.
T03.02 Determine, using a unit circle or reference triangle, the exact value of a trigonometric ratio for angles expressed in degrees that are multiples of 0°, 30°, 45°, 60°, or 90°, or for angles expressed in radians that are multiples of 0, π /6, π /4, π /3, or π /2 and explain the strategy.
T03.03 Determine, with or without technology, the measures, in degrees or radians, of the angles in a specified domain, given the value of a trigonometric ratio.
T03.04 Explain how to determine the exact values of the six trigonometric ratios, given the coordinates of a point on the terminal arm of an angle in standard position.
T03.05 Determine the measures of the angles in a specified domain in degrees or radians, given a point on the terminal arm of an angle in standard position.
T03.06 Determine the exact values of the other trigonometric ratios, given the value of one trigonometric ratio in a specified domain.
T03.07 Sketch a diagram to represent a problem that involves trigonometric ratios.
T03.08 Solve a problem, using trigonometric ratios.
T04 Students will be expected to graph and analyze the trigonometric functions sine, cosine, and tangent to solve problems. [CN, PS, T, V]
T04.01 Sketch, with or without technology, the graph of y = sin x, y = cos x, or y = tan x.
T04.02 Determine the characteristics (amplitude, asymptotes, domain, period, range, and zeros) of the graph of y = sin x, y = cos x, or y = tan x.
T04.03 Determine how varying the value of a affects the graphs of y = a sin x and y = a cos x.
T04.04 Determine how varying the value of d affects the graphs of y = sin x + d and y = cos x + d.
T04.05 Determine how varying the value of c affects the graphs of y = sin (x + c) and y = cos (x + c).
T04.06 Determine how varying the value of b affects the graphs of y = sin bx and y = cos bx.
T04.07 Sketch, without technology, graphs of the form y = a sin b(x − c) + d or y = a cos b(x − c) + d, using transformations, and explain the strategies.
T04.08 Determine the characteristics (amplitude, asymptotes, domain, period, phase shift, range and zeros) of the graph of a trigonometric function of the form y = a sin b(x − c) + d or y = a cos b(x − c) + d.
T04.09 Determine the values of a, b, c, and d for functions of the form y = a sin b(x − c) + d or y = a cos b(x − c) + d that correspond to a given graph, and write the equation of the function.
T04.10 Determine a trigonometric function that models a situation to solve a problem.
T04.11 Explain how the characteristics of the graph of a trigonometric function relate to the conditions in a problem situation.
T04.12 Solve a problem by analyzing the graph of a trigonometric function.
T05 Students will be expected to solve, algebraically and graphically, first- and second-degree trigonometric equations with the domain expressed in degrees and radians. [CN, PS, R, T, V]
T05.01 Verify, with or without technology, that a given value is a solution to a trigonometric equation.
T05.02 Determine, algebraically, the solution of a trigonometric equation, stating the solution in exact form, when possible.
T05.03 Determine, using technology, the approximate solution of a trigonometric equation in a restricted domain.
T05.04 Relate the general solution of a trigonometric equation to the zeros of the corresponding trigonometric function (restricted to sine and cosine functions).
T05.05 Determine, using technology, the general solution of a given trigonometric equation.
T05.06 Identify and correct errors in a solution for a trigonometric equation.
T06 Students will be expected to prove trigonometric identities, using
reciprocal identities
quotient identities
Pythagorean identities
sum or difference identities (restricted to sine, cosine, and tangent)
double-angle identities (restricted to sine, cosine, and tangent)
[R, T, V]
T06.01 Explain the difference between a trigonometric identity and a trigonometric equation.
T06.02 Verify a trigonometric identity numerically for a given value in either degrees or radians.
T06.03 Explain why verifying that the two sides of a trigonometric identity are equal for given values is insufficient to conclude that the identity is valid.
T06.04 Determine, graphically, the potential validity of a trigonometric identity, using technology.
T06.05 Determine the non-permissible values of a trigonometric identity.
T06.06 Prove, algebraically, that a trigonometric identity is valid.
T06.07 Determine, using the sum, difference, and double-angle identities, the exact value of a trigonometric ratio.
Additional Resources and Activities for T03 (trig ratios):
Exact Values Match Up - Students complete the values in this table to compare trig functions in degrees, radians and their exact values. Activity from Starting Points Maths.
Trig War - The classic card game of War played with trig unit circle expressions to review trig ratios. Trig war cards to print. Created by Sam Shah.
Trigonometry Duel - Unit Circle Desmos Activity - Students take the role of the referee. Each competitor is dealt a card with a trig ratio expression with a special angle from the unit circle. The referee determines which competitor has the expression with the higher value. You might let students use a printed unit circle as a reference until they are ready to play from memory.
Additional Resources and Activities for T04 (graph and analyze the trigonometric functions sine, cosine, and tangent):
Graphing Sin(x) and Cos(x) - Using string, unit circle and spaghetti to create graphs of sin(x) and cos(x). There is a great video of how this is created on TEDEd.
Graph Me If You Can - Sinusoidal Functions - Students work in small groups to graph a sinusoidal function from a verbal description only. Several different ways to facilitate this activity are suggested. Instructions are provided on this Google Slides as well as graphs to print off for students.
"Oops, I Forgot..." Sinusoidal Functions task - Ask students to write the equation for any sinusoidal function that passes through the point (𝜋, 2). Afterwards, add additional constraints one at a time such as, "the function must have an amplitude of 3". Read more about this routine on Nat Banting's blog.
Constructing the Sine and Cosine Graph using a Clothesline Number Line - Check out clothesline math resources.
Desmos Marbleslides: Periodics activity - In this delightful and challenging activity, students will transform periodic functions so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.
Additional Resources and Activities for T05 (solving trig equations):
Solving Trig Equation (in degrees) Tarsia Puzzle - Students cut out triangular puzzles pieces and assemble them so that sides with matching equations and solutions go together. These are introductory trig equations with all answers in degrees. Here is the solution and the Tarsia file if you'd like to edit it to add more difficult questions or to add solutions in radians.
Solving Trig Equations Secret Message - On this worksheet, students use trig identities to solve 6 equations. The answers are used to create a keyword. This keyword can then be used to decrypt a secret message that was created with a Vigenere cipher.
Additional Resources and Activities for T06 (prove trigonometric identities, using reciprocal identities, quotient identities, Pythagorean identities, sum or difference identities and double-angle identities):
Introduction to Trig Identities Desmos Activity - An introduction what what an identity is and the reciprocal, quotient and Pythagorean identities.
Desmos Activity: Sum, Difference, and Double Angle Identities *Updated Sept 2025* A Pre-calculus 12 activity for developing the opposite angle, compound angle, and double-angle identities for sine and cosine.
One or Negative One? - Five different trig identities each that simplify to either 1 or -1. Ask students to identify which are which. You could also pass out these questions as a handout. From Sarah Carter.
Trig Identity Match Up Activity - Students cut up a sheet containing statements from several trig identities. They then try to match up pairs of statements that are one step apart. They continue matching up statements until they have 4 complete trig identities. Created by Shireen Dadmehr.
Trig Identities Apartment Building - This Google slides is in the format of a "choose your own adventure" book. Students take the elevator from floor to floor and from room to room simplifying trig identities in order to find the correct path to the exit (similar to an escape room). Created by Girl Math.
Trig Sum Formulas - A visual to show the trig sum formula. How might you use this with students? Could they create it for themselves? Perhaps a Desmos Activity could help lead them in discovering this. Created by Joel Fish.
Double Angle Demo in Desmos - A nice visualization for the proof of sin(2a) = 2sin(a)cos(a) using the area of two congruent triangles.
Inspiration is for Amateurs - This is a quote from Chuck Close that can inspire students to start trying things when solving trig identities. "The advice I like to give young artists [or mathematicians], or really anybody who’ll listen to me, is not to wait around for inspiration. Inspiration is for amateurs; the rest of us just show up and get to work. If you wait around for the clouds to part and a bolt of lightning to strike you in the brain, you are not going to make an awful lot of work. All the best ideas come out of the process; they come out of the work itself. Things occur to you. If you’re sitting around trying to dream up a great art idea, you can sit there a long time before anything happens. But if you just get to work, something will occur to you and something else will occur to you and something else that you reject will push you in another direction. Inspiration is absolutely unnecessary and somehow deceptive. You feel like you need this great idea before you can get down to work, and I find that’s almost never the case."
Unit 4 Cumulative Review
Math Market Trigonometry Review - Student buy questions from the market and sell solutions back for a profit. A fun activity to review of a variety of trig outcomes including solving equations and proving identities. A recording sheet for student work can be helpful.