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Unit 7: Trigonometric Identities and Equations

The identities that let one trig expression stand in for another, how to prove them, and how to use them to solve equations by hand.

12 topics · 46 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

The Pythagorean Identities

  1. How it worksHow does the Pythagorean theorem on the unit circle turn into the identity sin2(x) + cos2(x) = 1?
  2. Worked exampleFind cos(x) given that sin(x) = 3/5 and x is in the second quadrant.
  3. Common mistakeWhy doesn't sin2(x) mean sin(x2)?
  4. Ties togetherHow does the Pythagorean identity let cos(2x) be rewritten in three equivalent double-angle forms?
Your turn1 of 2
Given sin(x) = a/8 (0 < a < 8) and x is in the second quadrant, find cos(x), as a function of a.

Even-Odd and Cofunction Identities

  1. How it worksHow does the unit circle's symmetry across the x-axis produce the even-odd identities?
  2. Worked exampleFind sin(-π/6) and cos(-π/6) using the even-odd identities.
  3. Common mistakeWhy does sin(π/2 - x) equal cos(x) instead of just equaling sin(x) minus something?
Your turn1 of 2
Find sin(-4pi/6).

Simplifying Trigonometric Expressions

  1. How it worksHow do you rewrite every trig function in an expression in terms of sine and cosine to simplify it?
  2. Worked exampleFind the simplified form of (sin(x)·cot(x))/cos(x).
  3. Worked exampleFind the simplified form of (1 - cos2(x))/sin(x).
  4. Common mistakeWhy can't you cancel the x in sin(x)/x the way you'd cancel a variable in an algebraic fraction?
Your turn1 of 2
Find the simplified form of (1 - cos2(x))/sin(x).

Proving Trigonometric Identities

  1. How it worksHow do you pick a starting side and rewrite it step by step until it matches the other side exactly?
  2. Worked exampleProve that (1 - cos(x))/sin(x) equals sin(x)/(1 + cos(x)).
  3. Worked exampleProve that sec(x) - cos(x) equals sin(x)·tan(x).
  4. Common mistakeWhy isn't squaring both sides of a trig identity a safe way to prove it's true?
Your turn1 of 2
Evaluate the left side of (1 - cos(x))/sin(x) = sin(x)/(1 + cos(x)) at x = 3pi/8.

Sum and Difference Formulas for Sine and Cosine

  1. How it worksHow does the sum formula cos(a+b) = cos(a)cos(b) - sin(a)sin(b) come from rotating one angle past another?
  2. Worked exampleFind the exact value of cos(75 degrees) using the sum formula with 45 and 30 degrees.
  3. Worked exampleFind the exact value of sin(105 degrees) using the sum formula with 60 and 45 degrees.
  4. Common mistakeWhy isn't sin(a+b) equal to sin(a) + sin(b)?
Your turn1 of 2
Write cos(x + 59 degrees) using the sum formula cos(a+b) = cos(a)cos(b) - sin(a)sin(b), in terms of cos(x) and sin(x).

Sum and Difference Formulas for Tangent

  1. How it worksHow does dividing the sine and cosine sum formulas produce the tangent sum formula?
  2. Worked exampleFind the exact value of tan(15 degrees) using the difference formula with 45 and 30 degrees.
  3. Common mistakeWhy doesn't tan(a+b) simplify to tan(a) + tan(b)?
Your turn1 of 2
Write tan(x - 19 degrees) using the difference formula, in terms of tan(x).

Double-Angle Formulas

  1. How it worksHow do the sum formulas collapse into the double-angle formulas when you set b equal to a?
  2. Worked exampleFind sin(2x) and cos(2x) given that sin(x) = 4/5 and x is in the first quadrant.
  3. Worked exampleFind tan(2x) given that tan(x) = 3/4.
  4. Common mistakeWhy isn't cos(2x) the same as 2·cos(x)?
Your turn1 of 2
Given sin(x) = a/8 (0 < a < 8) and x is in the first quadrant, find sin(2x), as a function of a.

Power-Reducing and Half-Angle Formulas

  1. How it worksHow do the double-angle formulas rearrange into the power-reducing formulas for sin2(x) and cos2(x)?
  2. Worked exampleFind the exact value of cos(15 degrees) using a half-angle formula.
  3. Common mistakeWhy does the half-angle formula for sine need a plus-or-minus sign in front of the square root?
  4. Ties togetherHow do the half-angle formulas come from solving the double-angle formulas backward?
Your turn1 of 2
Let u = cos(theta). Write 4·cos(theta/2) using the half-angle formula, in terms of u.

Product-to-Sum and Sum-to-Product Formulas

  1. How it worksHow do adding and subtracting the sum and difference formulas produce the product-to-sum identities?
  2. Worked exampleFind the product-to-sum form of sin(4x)·cos(2x).
  3. Worked exampleFind the sum-to-product form of sin(5x) + sin(3x).
  4. Going furtherWhat happens to a sum-to-product identity when the two angles being added are equal?
Your turn1 of 2
Write sin(x)·cos(0.927) using the product-to-sum identity, as a function of x.

Solving Trigonometric Equations by Factoring

  1. How it worksHow does setting each factor of a factored trig equation to zero find all of its solutions?
  2. Worked exampleFind all solutions to 2·sin2(x) - sin(x) - 1 = 0 on the interval from 0 to 2pi.
  3. Common mistakeWhy can't you divide both sides of sin(x)·cos(x) = sin(x) by sin(x) without losing solutions?
Your turn1 of 2
Find the smallest positive solution to (sin(x) - 0.657)·(sin(x) + 0.619) = 0 on the interval [0, π/2].

Solving Equations with Multiple Angles

  1. How it worksHow do you solve for the multiple angle first and then divide to isolate x, without dropping solutions?
  2. Worked exampleFind all solutions to sin(2x) = √3/2 on the interval from 0 to 2pi.
  3. Worked exampleFind all solutions to cos(3x) = 0 on the interval from 0 to 2pi.
  4. Common mistakeWhy does forgetting to add multiples of the period before dividing by 2 lose half the solutions to sin(2x) = k?
Your turn1 of 2
Find the smallest positive solution to sin(5x) = 0.619.

Solving Trigonometric Equations Using Identities

  1. How it worksHow does substituting the Pythagorean identity turn a mixed sine-cosine equation into one you can factor?
  2. Worked exampleFind all solutions to cos(2x) + sin(x) = 0 on the interval from 0 to 2pi using a double-angle identity.
  3. Worked exampleFind all solutions to 2·cos2(x) + sin(x) - 1 = 0 on the interval from 0 to 2pi using the Pythagorean identity.
  4. Common mistakeWhy can't you solve cos(2x) = sin(x) by treating 2x and x as if they cancel out?
  5. Going furtherWhat happens to the strategy for solving a trig equation when three different trig functions appear in it?
Your turn1 of 2
Find the solution in [0, π/2] to cos(2x) + 0.757·sin(x) = 0, using the double-angle identity cos(2x) = 1 - 2sin2(x).