Math Vale

🏔️ Trigonometry Peaks · Trigonometry

Trigonometric Identities

Use the Pythagorean and quotient identities to simplify expressions and verify equations.

In short

  • An identity holds for every angle, unlike an equation, which holds only for particular ones.
  • sin2(t) + cos2(t) = 1 is the equation of the unit circle, and its rearrangements are swap rules.
  • The identity gives the size of a ratio; only the quadrant can give the sign.
  • Cofunction identities pair angles that add to 90°: sin(t) = cos(90° - t).
  • The sum and difference formulas turn an unfamiliar angle into two standard ones, which is how exact values such as sin(75°) are found.
  • Sine mixes the two functions and keeps the sign in the bracket; cosine pairs like with like and reverses it.
  • sin(2t) = 2sin(t)cos(t), and cos(2t) has three interchangeable forms — pick the one built from the ratio you were given.

What an identity is

An equation is true for particular values: sin(t) = 1/2 holds only at certain angles. An identity is true for every value it makes sense for.

The most important one is the Pythagorean identity:

sin2(t) + cos2(t) = 1

The notation sin2(t) means (sin t)2 — square the ratio, not the angle.

It is not a new fact at all. On the unit circle the point at angle t is (cos t, sin t), and that point is 1 unit from the origin, so x2 + y2 = 1. The identity is the equation of the circle.

Because it holds for every angle, sin2(80°) + cos2(80°) = 1 without any calculation, and so does sin2(t) + cos2(t) for any t at all.

Rearranging the Pythagorean identity

The identity is most useful in its rearranged forms:

sin2(t) = 1 - cos2(t) cos2(t) = 1 - sin2(t)

Read them as swap rules. Whenever 1 - cos2 appears anywhere in an expression, it can be replaced by sin2, and vice versa — usually opening up a cancellation.

(1 - cos2(t)) / sin(t) = sin2(t) / sin(t) = sin(t)

Along with the definition tan(t) = sin(t)/cos(t), that is enough to simplify a great many expressions.

Note carefully which one is which. Subtracting sine squared from 1 leaves cosine squared. Getting that backwards is the commonest error in the topic.

From one ratio to another

Given one ratio and the quadrant, every other ratio can be recovered. The identity supplies the size and the quadrant supplies the sign.

"sin(t) = 3/5 and t is in the second quadrant. Find cos(t)."

cos2(t) = 1 - (3/5)2 = 1 - 9/25 = 16/25 so the size of cos(t) is 4/5 in quadrant 2 the x-coordinate is negative, so cos(t) = -4/5

The pattern is always: square, subtract from 1, take the square root, then fix the sign from the quadrant.

A quicker route for the neat cases is to draw a reference right triangle. sin = 3/5 means opposite 3 and hypotenuse 5, so the adjacent side is 4 by the Pythagorean theorem — and the triples 3-4-5, 5-12-13 and 8-15-17 show up again and again.

The square root step is where the sign is decided, and the identity alone can never tell you which one to take. Only the quadrant can.

Cofunction identities

In a right triangle the two acute angles add to 90°, and the side opposite one of them is adjacent to the other. So the sine of one angle equals the cosine of the other:

sin(t) = cos(90° - t) cos(t) = sin(90° - t)

That is what the "co-" in cosine means: the ratio of the complementary angle. So sin(25°) = cos(65°), and cos(10°) = sin(80°).

Note that cofunctions pair angles that add to 90°, not 180°. And except at 45°, sin(t) and cos(t) are not equal — the identity swaps the function as well as the angle.

Every identity gives a way to check work. If a simplification is right, both sides must agree at any convenient angle. Pick t = 30° and evaluate both — a numerical mismatch means an algebraic mistake, and it is often the fastest way to find it.

Sums, differences and double angles

The identities above relate ratios of the same angle. The next family relates the ratios of a combined angle.

sin(a + b) = sin(a)cos(b) + cos(a)sin(b) sin(a - b) = sin(a)cos(b) - cos(a)sin(b) cos(a + b) = cos(a)cos(b) - sin(a)sin(b) cos(a - b) = cos(a)cos(b) + sin(a)sin(b) tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

Two habits keep them straight. Sine mixes: every one of its terms has one sine and one cosine. Cosine matches: every one of its terms pairs like with like. And for the signs — sine keeps the sign that is inside the bracket, cosine reverses it. The tangent formula reverses it underneath.

Where cos(a - b) comes from, in a sketch: put two points on the unit circle at angles a and b. The distance between them can be found with the distance formula, or with the cosine rule on the triangle they make with the origin. Setting those two answers equal and tidying up leaves exactly cos(a)cos(b) + sin(a)sin(b). Everything else in the family follows from that one, by replacing b with -b or by a cofunction.

What the family is worth is exact values at angles the unit circle does not carry. Since 75° = 45° + 30°:

sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°) = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6) + sqrt(2))/4

Double angles are the same formulas with b = a:

sin(2t) = 2sin(t)cos(t) cos(2t) = cos2(t) - sin2(t) = 2cos2(t) - 1 = 1 - 2sin2(t) tan(2t) = sin(2t)/cos(2t)

The three forms of cos(2t) are one expression with the Pythagorean identity swapped in, and choosing the convenient one is most of the skill: given only a cosine, reach for 2cos2(t) - 1. Note that sin(2t) is not 2sin(t) — doubling an angle is not doubling its sine, and the quickest way to see it is that 2sin(t) can exceed 1 while a sine never does.

Typing your answer. Write roots as sqrt(...), so sin(75°) is typed (sqrt(6) + sqrt(2))/4 — an exact form, never a rounded decimal. Ratios that come out rational are typed as fractions in lowest terms, such as -24/25, and a shifted angle that collapses to one ratio is typed sin(t), cos(t), -sin(t) or -cos(t).

Worked examples

Example 1

Simplify (1 - sin2(t)) / cos(t).

  1. Rearrange the Pythagorean identity: 1 - sin2(t) = cos2(t).
  2. The expression becomes cos2(t) / cos(t).
  3. Cancel one factor of cos(t) from the top and the bottom.
  4. The expression simplifies to cos(t).

Example 2

cos(t) = 12/13 and t is in the fourth quadrant. Find sin(t).

  1. Use sin2(t) = 1 - cos2(t) = 1 - 144/169 = 25/169.
  2. Taking the square root, the size of sin(t) is 5/13.
  3. In the fourth quadrant the y-coordinate is negative, and sine is the y-coordinate.
  4. So sin(t) = -5/13.

Example 3

Find the exact value of cos(15°).

  1. Write 15° with two standard angles: 15° = 45° - 30°.
  2. Use cos(a - b) = cos(a)cos(b) + sin(a)sin(b). Cosine reverses the sign, so a difference of angles gives a plus in the middle.
  3. cos(45°)cos(30°) + sin(45°)sin(30°) = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2)
  4. That is sqrt(6)/4 + sqrt(2)/4 = (sqrt(6) + sqrt(2))/4.
  5. Sense check: the value is about 0.966, and 15° is close to 0°, where the cosine is 1.

Example 4

sin(t) = 3/5 and t is in the second quadrant. Find sin(2t) and cos(2t).

  1. Both formulas need cos(t), so find it first: cos2(t) = 1 - 9/25 = 16/25, so the size of cos(t) is 4/5.
  2. In the second quadrant the cosine is negative, so cos(t) = -4/5.
  3. sin(2t) = 2sin(t)cos(t) = 2 * (3/5) * (-4/5) = -24/25.
  4. cos(2t) = cos2(t) - sin2(t) = 16/25 - 9/25 = 7/25.
  5. Sense check: sin(2t) is negative and cos(2t) is positive, so 2t sits in the fourth quadrant — and t is about 143°, so 2t is about 286°.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Simplify 1 - sin2(t). (The angle is written t.)

  1. sin2(t)
  2. cos(t)
  3. cos2(t)

Answer: C. cos2(t)

  1. Start with 1 - sin2(t).
  2. Rearranging sin2 + cos2 = 1 gives cos2(t) = 1 - sin2(t).
  3. It simplifies to cos2(t).

Problem 2

Difficulty 3 of 5

sin(t) = -3/5 and 270° < t < 360°. Find cos(t). Give your answer as a fraction.

Answer: 4/5

  1. sin2(t) = (-3/5)2 = 9/25
  2. cos2(t) = 1 - 9/25 = 16/25
  3. cos(t) = 4/5 (quadrant 4)

Problem 3

Difficulty 4 of 5

cos(63°) = sin(?) What angle, in degrees between 0° and 90°, goes in the box?

Answer: 27 degrees

  1. sin(t) = cos(90° - t) for any angle t.
  2. cos(63°) = sin(90° - 63°)
  3. 90 - 63 = 27°

Common mistakes

  • Rearranging the identity backwards, so 1 - sin2 becomes sin2 instead of cos2.
  • Reading sin2(t) as the sine of t squared rather than the square of sin(t).
  • Taking the positive square root automatically and ignoring the quadrant.
  • Using 180° instead of 90° in a cofunction identity.
  • Writing sin(a + b) as sin(a) + sin(b) — test it at 30° and 60° and the two sides disagree immediately.
  • Putting a plus in the middle of cos(a + b): cosine reverses the sign, so it is cos(a)cos(b) - sin(a)sin(b).
  • Writing sin(2t) as 2sin(t), or cos(2t) as cos2(t) + sin2(t), which is only ever 1.
  • Losing the sign of cos(t) in the second quadrant, so sin(2t) comes out positive when it should be negative.

What you should be able to do

  • Apply sin2 t + cos2 t = 1 and its rearrangements.
  • Use the quotient and reciprocal identities.
  • Simplify a trigonometric expression to a single function.
  • Verify an identity step by step.

Where this fits in the curriculum

Common Core

  • HSF-TF.C.9

    High school — Prove the addition and subtraction formulas for sine, cosine and tangent and use them to solve problems.

    HSF-TF.C.9 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II.

  • HSF-TF.C.8

    High school — Prove the Pythagorean identity sin²θ + cos²θ = 1 and use it to find sine, cosine or tangent given one of them and the quadrant.

  • HSF-TF.A.4

    High school — Use the unit circle to explain the symmetry and periodicity of the trigonometric functions.

    The quotient and reciprocal identities have no standard of their own; the Common Core treats them as definitions used inside HSF-TF.C.8. HSF-TF.A.4 is a (+) standard.

SAT

  • Additional Topics in Math

    The Pythagorean identity and complementary-angle relationships.

    Identities appear rarely on the SAT, and almost always as sin²+cos²=1 or sin(x) = cos(90° − x).

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