Trigonometry

Sine and cosine defined as coordinates on the unit circle, the exact values for the special angles obtained from triangle geometry, the identities derived from the circle's equation and the angle-addition formulas, and the equations whose solutions repeat forever.

Definition

The unit circle definition. For a real number $θ$ , travel $θ$ units counterclockwise along the unit circle $x^{2} + y^{2} = 1$ from the point $(1, 0)$ . The endpoint has coordinates

(\cos θ, \sin θ) .

Cosine is the first coordinate and sine the second. The other functions are defined from these: $\tan θ = \frac{\sin θ}{\cos θ}$ , with $\sec$ , $\csc$ and $\cot$ the reciprocals of cosine, sine and tangent.

Radians. $θ$ is an arc length, so the natural unit is the radian: a full revolution is $2 π$ , a half is $π$ , a quarter is $π / 2$ . Degrees convert by $π$ radians $= 180 °$ .

Domain and range. Sine and cosine are defined for every real $θ$ and take values in $[- 1, 1]$ , since the coordinates of a point on the unit circle cannot exceed 1 in magnitude. Tangent is undefined wherever $\cos θ = 0$ , that is at $θ = π / 2 + k π$ .

Periodicity. Travelling a further $2 π$ returns to the same point, so $\sin (θ + 2 π) = \sin θ$ and likewise for cosine. Tangent has period $π$ rather than $2 π$ .

Special angles. Two triangles supply the exact values. The 45-45-90 triangle has legs $1, 1$ and hypotenuse $\sqrt{2}$ ; the 30-60-90 has sides $1, \sqrt{3}, 2$ . Reading the ratios gives:

$θ$	$0$	$π / 6$	$π / 4$	$π / 3$	$π / 2$
$\cos θ$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\sin θ$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$

The Pythagorean identity. The definition places $(\cos θ, \sin θ)$ on $x^{2} + y^{2} = 1$ , so

\cos^{2} θ + \sin^{2} θ = 1

for every $θ$ . It is the circle's equation restated, not a separate fact. Dividing through by $\cos^{2} θ$ or $\sin^{2} θ$ gives $1 + \tan^{2} θ = \sec^{2} θ$ and $1 + \cot^{2} θ = \csc^{2} θ$ .

Angle addition. For all $α, β$ :

\sin (α + β) = \sin α \cos β + \cos α \sin β, \cos (α + β) = \cos α \cos β - \sin α \sin β .

Setting $β = α$ gives the double-angle formulas $\sin 2 α = 2 \sin α \cos α$ and $\cos 2 α = \cos^{2} α - \sin^{2} α$ , so those are consequences rather than additions to the list.

Solving trigonometric equations. Because the functions are periodic, an equation with one solution has infinitely many. The method is to find every solution in one period, then add the period's multiples: $\sin θ = c$ has solutions $θ_{0} + 2 k π$ and $(π - θ_{0}) + 2 k π$ for integer $k$ , where $θ_{0} = \arcsin c$ .

Assumptions and scope

The unit circle definition covers every real $θ$ ; the right-triangle definition covers only acute angles and cannot express $\cos (2 π / 3)$ at all.
Radians are the default in analysis. The derivative formulas $\frac{d}{d x} \sin x = \cos x$ and the limit $lim_{x \to 0} \frac{\sin x}{x} = 1$ are false if $x$ is measured in degrees.
$\tan θ$ is undefined wherever $\cos θ = 0$ , at $θ = π / 2 + k π$ . The function has no value there, rather than a large one.
Sine and cosine take values only in $[- 1, 1]$ , so an equation such as $\sin θ = 2$ has no solution at all.
A trigonometric equation has infinitely many solutions unless an interval is specified. Reporting a single value answers a different question from the one asked.
$\arcsin$ and $\arccos$ return one value from a restricted range, not every solution. Recovering the rest requires the symmetry of the circle.

Forms this is expressed in

The same content in several forms. Each makes something visible that the others leave implicit, so moving between them is part of understanding the topic rather than a presentation choice.

geometric

The unit circle, with θ chosen by the reader

The circle $x^{2} + y^{2} = 1$ with an angle $θ$ measured counterclockwise from $(1, 0)$ , and the endpoint labelled $(\cos θ, \sin θ)$ .

This form makes the definitions and their consequences the same picture. The coordinates cannot exceed 1 in magnitude, so the range $[- 1, 1]$ is visible rather than stated. A full revolution returns to the starting point, so periodicity is the journey closing. And the point lying on the circle is the Pythagorean identity: $\cos^{2} θ + \sin^{2} θ = 1$ is the defining equation with the coordinates renamed.

Signs by quadrant are read off directly. In the second quadrant the first coordinate is negative and the second positive, so $\cos (2 π / 3) = - \frac{1}{2}$ while $\sin (2 π / 3) = + \frac{\sqrt{3}}{2}$ . No sign rule needs remembering, because the position supplies it.

The circle also shows why an equation has infinitely many solutions and where they sit. $\sin θ = \frac{1}{2}$ means the second coordinate is $\frac{1}{2}$ , and a horizontal line at that height crosses the circle twice, at $π / 6$ and $5 π / 6$ : the two solutions in one revolution, repeating every $2 π$ thereafter. The reflection symmetry about the vertical axis is why the second solution is $π - θ_{0}$ .

What this form cannot supply is an exact value. The picture shows that $\cos (π / 6)$ is somewhat more than $0.8$ ; that it is exactly $\frac{\sqrt{3}}{2}$ comes from the 30-60-90 triangle, not from the circle.

Translates into: graphical

graphical

Sine and cosine over one revolution, with the line y = ½

The same functions plotted against $θ$ on a horizontal axis: $y = \sin θ$ and $y = \cos θ$ as waves oscillating between $- 1$ and $1$ with period $2 π$ , and $y = \tan θ$ with vertical asymptotes at $π / 2 + k π$ .

This form makes behaviour over an interval visible where the circle shows one angle at a time. Periodicity appears as the pattern repeating; the bound $[- 1, 1]$ appears as two horizontal lines the waves touch but never cross; and the phase relationship is legible at a glance, since $\cos$ is $\sin$ shifted left by $π / 2$ .

Solving an equation becomes an intersection count. The line $y = \frac{1}{2}$ meets $y = \sin θ$ twice per period, which is why $\sin θ = \frac{1}{2}$ has two solutions in $[0, 2 π)$ and infinitely many overall. The line $y = 2$ meets it never, which is why $\sin θ = 2$ has no solution, a fact the graph shows immediately.

Tangent's graph shows what its formula asserts: the curve climbs without bound as $θ$ approaches $π / 2$ from below, passing $10^{4}$ at $1.5707$ and $3 \times 10^{6}$ at $1.570796$ . The asymptote marks an angle where the function has no value, not a large one, and the period is $π$ rather than $2 π$ because the pattern repeats after half a revolution.

What this form loses is the reason behind the values. The graph shows that sine is bounded; the circle explains why: it is a coordinate of a point at distance 1 from the origin. Exact values are equally invisible here, requiring the reference triangles.

Translates into: geometric

Worked material

Example

The angles worth knowing on sight

The first-quadrant table. Every entry comes from one of the two reference triangles:

$θ$	$0$	$π / 6$	$π / 4$	$π / 3$	$π / 2$
$\cos θ$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\sin θ$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$

Each verified to ten decimal places against the circle. Cosine descends while sine climbs, which is the point travelling counterclockwise from $(1, 0)$ to $(0, 1)$ .

$θ = 2 π / 3$ (second quadrant). Reference angle $π / 3$ , first coordinate negative: $\cos = - \frac{1}{2}$ , $\sin = + \frac{\sqrt{3}}{2}$ . The magnitudes repeat from the first quadrant; only the signs change.

$θ = 7 π / 6$ (third quadrant). Reference angle $π / 6$ , both coordinates negative: $\cos = - \frac{\sqrt{3}}{2}$ , $\sin = - \frac{1}{2}$ . Check: $\sin (7 π / 6) = - 0.5000000000$ .

$θ = 11 π / 6$ (fourth quadrant). Reference angle $π / 6$ , first coordinate positive and second negative: $\cos = + \frac{\sqrt{3}}{2}$ , $\sin = - \frac{1}{2}$ . Check: $\sin (11 π / 6) = - 0.5000000000$ .

These two share a sine and differ in cosine, which is why $\sin θ = - \frac{1}{2}$ has exactly these two solutions per revolution.

$θ = π / 12$ , from $45 ° - 30 °$ . Not a special angle itself, but a difference of two:

\sin \frac{π}{12} = \sin 45 ° \cos 30 ° - \cos 45 ° \sin 30 ° = \frac{\sqrt{6} - \sqrt{2}}{4} \approx 0.258819045103 .

Check: $\sin (π / 12) = 0.258819045103$ . Angle addition extends the table indefinitely.

$θ = π / 2$ for tangent. Undefined, because $\cos (π / 2) = 0$ and the quotient divides by it. The computed cosine is $6.1 \times 10^{- 17}$ , zero to machine precision, and the tangent values nearby climb $10,381$ , $158,058$ , $3,060,023$ at $1.5707$ , $1.57079$ , $1.570796$ . The function has no value there, and the same holds at every $π / 2 + k π$ .

$θ$ beyond a revolution: $\frac{13 π}{6}$ . Subtract $2 π$ to get $π / 6$ , the same point on the circle, so the values are identical: $\sin = \frac{1}{2}$ , $\cos = \frac{\sqrt{3}}{2}$ . Periodicity means any angle can be reduced to $[0, 2 π)$ before evaluating.

Four quadrants supply four sign patterns for the same magnitudes, angle addition reaches the gaps between special angles, and periodicity reduces everything else to one revolution. The table is five columns; the reachable values are unlimited.

Non-example

Notation and reasoning that go wrong

$\sin^{2} θ$ is not $\sin (θ^{2})$ . The notation abbreviates $(\sin θ)^{2}$ : the value squared, not the angle. At $θ = 2$ :

(\sin 2)^{2} \approx 0.8268218104, \sin (4) \approx - 0.7568024953 .

Different numbers, and different signs. Read under the wrong convention, the Pythagorean identity would claim $\sin (θ^{2}) + \cos (θ^{2}) = 1$ , which fails at once: at $θ = 2$ that sum is $\sin 4 + \cos 4 \approx - 1.4104$ , not 1.

A reference angle is not the answer. For $\cos (2 π / 3)$ the reference angle is $π / 3$ with $\cos (π / 3) = \frac{1}{2}$ , but the angle lies in the second quadrant where first coordinates are negative. The value is $- \frac{1}{2}$ . Reporting $+ \frac{1}{2}$ uses the magnitude and discards the position.

The Pythagorean identity does not fix a sign. Given $\cos t = \frac{3}{5}$ , the identity gives $\sin^{2} t = \frac{16}{25}$ , so $\sin t = \pm \frac{4}{5}$ . Both are consistent with it, one in the first quadrant and one in the fourth. Choosing requires the extra condition, and a problem stating none has two answers rather than one.

$\tan (π / 2)$ is undefined, not infinite. The quotient $\sin / \cos$ divides by $\cos (π / 2) = 0$ . The nearby values grow without bound, reaching $10,381$ at $1.5707$ and $3,060,023$ at $1.570796$ , but the function has no value at $π / 2$ . Writing $\tan (π / 2) = \infty$ names a number that is not one.

Reporting one solution to a periodic equation. $\sin θ = \frac{1}{2}$ has solutions $π / 6$ and $5 π / 6$ in $[0, 2 π)$ , and infinitely many overall. Giving $π / 6$ alone omits the second-quadrant solution, which is the most common error in solving these, since the calculator's $\arcsin$ returns one value from a restricted range and says nothing about the rest.

An equation with no solution. $\sin θ = 2$ has none, because no point on the unit circle has second coordinate 2. Applying $\arcsin$ regardless produces an error rather than an answer, and the check $| c | \leq 1$ catches it before any work is done.

Degrees where radians are required. $\frac{d}{d x} \sin x = \cos x$ is false for $x$ in degrees; the correct derivative carries a factor of $π / 180 \approx 0.01745$ . The same constant spoils $lim_{x \to 0} \frac{\sin x}{x} = 1$ , which becomes $0.01745$ . Analysis uses radians for exactly this reason, and a degree-mode calculator silently produces wrong derivatives.

Proving an identity by working both sides. Transforming the left and right simultaneously until they meet can establish a false statement, since the steps need not be reversible. The sound method transforms one side into the other.

Common errors

Common misconception

$\sin^{2} θ$ means $\sin (θ^{2})$ , so the Pythagorean identity reads $\sin (θ^{2}) + \cos (θ^{2}) = 1$ .

Common misconception

Exact trigonometric values exist only at the tabulated special angles, so an angle such as $\frac{7 π}{12}$ admits nothing better than a decimal approximation.

Trigonometry

Assumptions and scope

Forms this is expressed in

Worked material

The angles worth knowing on sight

Notation and reasoning that go wrong

Common errors

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