MHF4U Basic Trigonometric Identities: Notes and Practice

You already know the six trig ratios and the unit circle. What you may not have noticed is that those six ratios are not six independent things — they are tangled together by a handful of relationships that hold for EVERY angle, not just special ones. Those relationships are called identities, and they are the tools you will use for the rest of this unit: to rewrite messy expressions, to prove statements, and to solve equations that look impossible at first glance.

Identity versus equation

An IDENTITY is a statement that is true for every value of the variable for which both sides are defined. An EQUATION is a statement that happens to be true for particular values, and your job is to hunt those values down. The same symbols ( $=$ , $sin$ , $θ$ ) appear in both, so the only way to tell them apart is to ask what you are being asked to do: prove that something always holds, or find the values that make something hold.

AIdentity

Example: $sin^{2} θ + cos^{2} θ = 1$ .
True for every $θ$ in the domain.
Task word: PROVE or VERIFY.
Answer is an argument, not a number.
Restrictions matter: $tan θ = \frac{sin θ}{cos θ}$ fails where $cos θ = 0$ .

BEquation

Example: $sin θ = \frac{1}{2}$ .
True only for particular values of $θ$ .
Task word: SOLVE.
Answer is a list of angles.
The interval matters: on $0 \leq θ < 2 π$ that example has exactly two answers.

Reciprocal and quotient identities

The three reciprocal ratios are nothing more than the original three flipped upside down, and the two quotient identities say that tangent and cotangent are built out of sine and cosine. Together these five let you rewrite ANY trig expression using only $sin$ and $cos$ — which is the single most reliable first move in this entire unit.

csc θ = \frac{1}{sin θ} sec θ = \frac{1}{cos θ} cot θ = \frac{1}{tan θ} tan θ = \frac{sin θ}{cos θ} cot θ = \frac{cos θ}{sin θ}

Reciprocal identities (cosecant goes with sine, secant goes with cosine) and the two quotient identities.

The Pythagorean identities

On the unit circle the point at angle $θ$ is $(cos θ, sin θ)$ , and every point on that circle satisfies $x^{2} + y^{2} = 1$ . Substituting gives the first and most important identity of the course. (Note the notation: $sin^{2} θ$ means $(sin θ)^{2}$ , not $sin (θ^{2})$ .) You do not need to memorize the other two separately, because each is one division away from the first: divide every term of $sin^{2} θ + cos^{2} θ = 1$ by $cos^{2} θ$ and you get $tan^{2} θ + 1 = sec^{2} θ$ , or divide every term by $sin^{2} θ$ instead and you get $1 + cot^{2} θ = csc^{2} θ$ . Rearranged forms are just as useful: $sin^{2} θ = 1 - cos^{2} θ$ and $cos^{2} θ = 1 - sin^{2} θ$ are the substitutions you will reach for most often in a proof.

Fig 1The unit circle gives the first Pythagorean identity; dividing it by cosine squared and by sine squared gives the other two.

sin^{2} θ + cos^{2} θ = 1 ⟹ tan^{2} θ + 1 = sec^{2} θ ⟹ 1 + cot^{2} θ = csc^{2} θ

The three Pythagorean identities. Only the first is worth memorizing on its own.

VideoCovers the reciprocal, quotient and Pythagorean identities together and shows how they are used to rewrite trig expressions.Watch on YouTube

Even, odd, and cofunction identities

Reflecting an angle across the $x$ -axis (replacing $θ$ with $- θ$ ) leaves the $x$ -coordinate alone and flips the sign of the $y$ -coordinate. Since $x = cos θ$ and $y = sin θ$ , cosine is unchanged (it is an EVEN function) while sine flips sign (it is ODD), and tangent, being sine over cosine, is odd as well. The cofunction identities come from a different picture: the two acute angles of a right triangle add to $\frac{π}{2}$ , and what is opposite one of them is adjacent to the other, so subtracting an angle from $\frac{π}{2}$ turns each ratio into its co-ratio.

cos (- θ) = cos θ sin (- θ) = - sin θ tan (- θ) = - tan θ sin (\frac{π}{2} - θ) = cos θ cos (\frac{π}{2} - θ) = sin θ tan (\frac{π}{2} - θ) = cot θ

Top row: even and odd identities, with cosine the even one. Bottom row: the cofunction identities, which is literally where the co in cosine, cotangent and cosecant comes from.

Fig 2Every identity on this page can be read straight off the unit circle if you remember that the point is cosine across and sine up.

The default first move

When an expression contains secant, cosecant, cotangent or tangent and you have no idea what to do, convert everything to sine and cosine. It almost never makes things worse, and it very often makes the next step obvious.

Basic Trigonometric Identities

Key things to remember

Identity versus equation

Reciprocal and quotient identities

The Pythagorean identities

Even, odd, and cofunction identities

Worked examples

Where marks get lost

Flashcards 9

Practice