Contents / Precalculus / Trigonometric Functions

Chapter 4

Trigonometric Functions

Sine, cosine, tangent, the unit circle, and core trigonometric identities.

Introduction

Sine, cosine, tangent, the unit circle, and core trigonometric identities.

4.1Trigonometric functions

Trigonometric functions connect angles with ratios of lengths and are indispensable for describing periodic phenomena — waves, oscillations, circular motion, seasons, and alternating current.

Trigonometric Functions Overview

Definition 4.1 (The six trigonometric functions). For an acute angle $θ$ in a right triangle, the six trigonometric functions are ratios of sides:

sin θ = \frac{opposite}{hypotenuse}, cos θ = \frac{adjacent}{hypotenuse}, tan θ = \frac{opposite}{adjacent},

and the three reciprocals

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

The triangle only makes sense for an acute $θ$ , but these extend to all angles via the unit circle or reference angles — that extension is what turns six ratios into six functions of a real variable.

Proposition 4.2 (Periodicity). All six trigonometric functions are periodic: sine and cosine have period $2 π$ , and tangent has period $π$ .

Intuition. Think of a ramp. Its angle alone decides its shape: a steeper angle means the far end rises higher for the same length of ramp. Trig functions are the lookup table for that fact — feed in the angle and you get back the exact ratio of "how high" to "how far," no matter how big the actual ramp is. Sine, cosine, and tangent are just three different ratios of the ramp's sides.

Example 4.3 (Finding all six trig values for a 3-4-5 right triangle with acute angle $θ$ opposite the side of length 3). Label the sides relative to $θ$ : opposite $= 3$ , adjacent $= 4$ , hypotenuse $= 5$ (the hypotenuse is always the longest side, opposite the right angle).

Solution.

Primary ratios: $sin θ = \frac{opp}{hyp} = \frac{3}{5}$ , $cos θ = \frac{adj}{hyp} = \frac{4}{5}$ , $tan θ = \frac{opp}{adj} = \frac{3}{4}$ .
Reciprocals: $csc θ = \frac{1}{s i n θ} = \frac{5}{3}$ , $sec θ = \frac{1}{c o s θ} = \frac{5}{4}$ , $cot θ = \frac{1}{t a n θ} = \frac{4}{3}$ .
Sanity check with the Pythagorean identity: $sin^{2} θ + cos^{2} θ = (\frac{3}{5})^{2} + (\frac{4}{5})^{2} = \frac{9}{25} + \frac{16}{25} = \frac{25}{25} = 1$ . ✓ The values are consistent.

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Common Mistakes

Radians vs degrees: Calculators and most advanced math use radians by default. sin(30) in radians is not 0.5; always check mode.
Reference angle signs: Forgetting to apply the correct sign for sine/cosine/tangent in each quadrant leads to wrong signs.
Identity direction: sin² + cos² =1 can be used to find one from the other, but you must take ± square root and choose sign from quadrant info.
Undefined tangent: tan and sec are undefined at odd multiples of π/2; students sometimes plot them or solve equations as if defined there.
Unit circle memorization gaps: Not knowing the 30-45-60 exact values forces calculator use and loses exact answers on exams.
Forgetting the three reciprocal functions: csc sec cot are 1 over the primary three; useful in identities.

Exercises for §4.1 →

4.2Sine

Sine is the fundamental periodic function, giving the vertical position as you travel around the unit circle.

Sine

Definition 4.4 (Sine). The sine function $sin θ$ gives the $y$ -coordinate of the point on the unit circle reached by traveling an arc length $θ$ (in radians) counterclockwise from $(1, 0)$ .

Everything else about sine follows from that picture: the point never leaves the circle, and after a full lap it is back where it started.

Proposition 4.5 (Basic properties of sine). Sine is an odd function,

sin (- θ) = - sin θ,

so its graph has point symmetry about the origin. Its range is $[- 1, 1]$ , its period is $2 π$ , and its zeros are at integer multiples of $π$ .

Sine is the "vertical" component in circular motion and the imaginary part of $e^{i θ}$ (Euler's formula: $e^{i θ} = cos θ + i sin θ$ ).

Intuition. Sine tells you "how high or low you are" as you walk around a circle at constant speed. It starts at 0, goes up to 1 at 90°, back to 0 at 180°, down to -1, and repeats.

Example 4.6 (Graph the key points of $y = sin x$ over one period, then evaluate $sin (\frac{7 π}{6})$ ). Read the five "quarter points" off the unit circle by tracking the $y$ -coordinate as $x$ increases:

Solution.

$x = 0$ : $sin 0 = 0$ (starting at $(1, 0)$ ).
$x = \frac{π}{2}$ : $sin \frac{π}{2} = 1$ (top of the circle).
$x = π$ : $sin π = 0$ (left side).
$x = \frac{3 π}{2}$ : $sin \frac{3 π}{2} = - 1$ (bottom).
$x = 2 π$ : $sin 2 π = 0$ (back to start).
Connecting these smoothly gives one full wave oscillating between $- 1$ and $1$ with wavelength $2 π$ .
Now use the graph's shape to find $sin (\frac{7 π}{6})$ . Since $\frac{7 π}{6} = π + \frac{π}{6}$ lies just past $π$ (quadrant III), sine is negative there. The reference angle is $\frac{π}{6}$ , so $sin (\frac{7 π}{6}) = - sin \frac{π}{6} = - \frac{1}{2}$ .
Sanity check: at $x = π$ the wave crosses zero heading downward, so a value slightly past $π$ should be small and negative — and $- \frac{1}{2}$ is exactly that. ✓

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Common Mistakes

Radians vs degrees: Calculators and most advanced math use radians by default. sin(30) in radians is not 0.5; always check mode.
Sign errors in quadrants: sin positive in QI and QII, negative in QIII QIV.
Period of sin vs tan: sin period 2π; tan π.
Forgetting sin is odd: sin(-θ) = -sin θ; useful for symmetry.
Zeros at kπ not k π/2: At π/2 sin=1, not zero.
Range always [-1,1]: No matter the angle, |sin| ≤1.

Exercises for §4.2 →

4.3Cosine

Cosine gives the horizontal position on the unit circle. It is the even counterpart to sine and phase shifted from it.

Cosine

Definition 4.7 (Cosine). Cosine $cos θ$ is the $x$ -coordinate of the point on the unit circle after turning through angle $θ$ .

Proposition 4.8 (Basic properties of cosine). Cosine is even,

cos (- θ) = cos θ,

so its graph is mirror-symmetric across the $y$ -axis. Its range is $[- 1, 1]$ , its period is $2 π$ , and its zeros are at $\frac{π}{2} + kπ$ for integer $k$ .

Proposition 4.9 (Cosine is a phase-shifted sine). For every angle $θ$ ,

cos θ = sin (θ + \frac{π}{2}) .

Intuitively, the $x$ -coordinate at angle $θ$ equals the $y$ -coordinate a quarter-turn later, so the cosine wave is just the sine wave slid left by $\frac{π}{2}$ .

It models the horizontal component of circular motion and the real part of $e^{i θ}$ (Euler's formula).

Intuition. Cosine is "how far left or right you are" on the circle. It starts at 1 when angle=0, drops to 0 at 90°, to -1 at 180°, etc. Same wave as sine but shifted left by 90°.

Example 4.10 (Using cosine to find the adjacent side, then the opposite side). In a right triangle with hypotenuse $10$ and an acute angle of $60°$ , find the side adjacent to that angle.

Solution. By definition $cos 60° = \frac{adjacent}{hypotenuse}$ , and from the unit circle $cos 60° = \frac{1}{2}$ , so $\frac{1}{2} = \frac{adjacent}{10}$ . Solve: $adjacent = 10 \cdot \frac{1}{2} = 5$ . For the opposite side, $sin 60° = \frac{3}{2} = \frac{opposite}{10}$ , so $opposite = 10 \cdot \frac{3}{2} = 53$ . Sanity check with the Pythagorean theorem: $5^{2} + (53)^{2} = 25 + 75 = 100 = 1 0^{2}$ . ✓ The two legs and hypotenuse fit.□

Common Mistakes

Radians vs degrees: Always check calculator mode.
Even vs odd: cos even, sin and tan odd.
Reference angle signs: Cos positive in QI QIV, negative QII QIII.
Zeros at odd multiples of π/2: Not the same as sin.
Confusing with sin in identities: cos θ = sin( θ + π/2) not sin(θ + π/2) = cos.
Phase shift direction: cos θ = sin(θ + π/2) is shift left (advance).

Exercises for §4.3 →

4.4Tangent

Tangent is the slope of the line from origin to the unit circle point. It has period π and vertical asymptotes.

Tangent

Definition 4.11 (Tangent). Tangent is defined as

tan θ = \frac{sin θ}{cos θ},

which for an acute angle equals $\frac{opposite}{adjacent}$ .

Because it is a quotient, tangent is only defined where $cos θ \neq = 0$ , and that single restriction shapes the whole graph.

Proposition 4.12 (Basic properties of tangent). The period of tangent is $π$ (not $2 π$ ) and its range is all real numbers. It has vertical asymptotes at the odd multiples of $\frac{π}{2}$ (where $cos θ = 0$ ), and its zeros are at integer multiples of $π$ (where $sin θ = 0$ ).

The tangent function measures slopes (rise over run) and appears in many trig-substitution integrals.

Intuition. Tangent is the slope of the line from origin to the point on the circle. As you approach 90° from left the slope shoots to +∞; just after 90° it comes from -∞.

Example 4.13 (Finding $tan 135°$ ). Locate the angle: $135° = 180° - 45°$ , so it lies in quadrant II with reference angle $45°$ .

Solution.

Determine signs by quadrant: in quadrant II, sine is positive and cosine is negative.
Apply the reference angle with those signs: $sin 135° = + sin 45° = \frac{2}{2}$ and $cos 135° = - cos 45° = - \frac{2}{2}$ .
Take the quotient: $tan 135° = \frac{sin 135°}{cos 135°} = \frac{2 /2}{- 2 /2} = - 1$ .
Sanity check: tangent is negative in quadrant II, and the reference angle $45°$ has $tan 45° = 1$ , so the magnitude $1$ with a negative sign is exactly what we expect. ✓

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Common Mistakes

Radians vs degrees: Calculators and most advanced math use radians by default. sin(30) in radians is not 0.5; always check mode.
Undefined tangent: tan and sec are undefined at odd multiples of π/2; students sometimes plot them or solve equations as if defined there.
Period π not 2π: tan repeats every 180°, not 360°.
Sign in quadrants: tan positive QI QIII, negative QII QIV.
tan = opp/adj only for acute: For general angles use unit circle or reference with signs.
Confusing with arctan range: arctan range (-π/2,π/2) not the same as tan's behavior.

Exercises for §4.4 →

4.5Unit circle

The unit circle is the dictionary for all trig values. Memorizing the key angles gives exact answers without calculator.

The Unit Circle

Definition 4.14 (Unit circle). The unit circle is the circle of radius $1$ centered at the origin. Its points are

(cos θ, sin θ),

where $θ$ is the arc length — that is, the angle in radians — measured from the positive $x$ -axis.

Measuring the angle as an arc length is exactly what radian measure means, and it is what makes the coordinates of the circle be the cosine and sine.

Common angles and exact values:

θ (radians)	θ (degrees)	sin θ	cos θ	tan θ
0	0°	0	1	0
π/6	30°	1/2	√3/2	1/√3
π/4	45°	√2/2	√2/2	1
π/3	60°	√3/2	1/2	√3
π/2	90°	1	0	undef
π	180°	0	-1	0
3π/2	270°	-1	0	undef
2π	360°	0	1	0

Memorizing these (and signs by quadrant) lets you evaluate trig functions exactly without calculator.

Intuition. The unit circle is a "dictionary" that translates any angle directly into its sin and cos by walking the distance around the circle and reading the coordinates. Radians make the numbers match the actual arc length.

Example 4.15 (Exact value of $cos (\frac{5 π}{3})$ ). Convert to degrees to place it: $\frac{5 π}{3} = \frac{5}{3} \cdot 180° = 300°$ , which is in quadrant IV.

Solution.

Find the reference angle (the acute angle to the nearest $x$ -axis): $360° - 300° = 60°$ .
Determine the sign: cosine (the $x$ -coordinate) is positive in quadrant IV.
Combine: $cos (\frac{5 π}{3}) = + cos 60° = \frac{1}{2}$ .
Sanity check: $300°$ is close to $360°$ , where the point is near $(1, 0)$ , so $cos$ should be positive and fairly large — $\frac{1}{2}$ fits. ✓

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Example 4.16 (Coordinates of the point at $\frac{2 π}{3}$ on the unit circle). Convert: $\frac{2 π}{3} = 120°$ , which is in quadrant II.

Solution.

Reference angle: $180° - 120° = 60°$ .
Signs in quadrant II: $x = cos$ is negative, $y = sin$ is positive.
Read magnitudes from the table for $60°$ : $cos 60° = \frac{1}{2}$ , $sin 60° = \frac{3}{2}$ .
Apply signs: the point is $(- \frac{1}{2}, \frac{3}{2})$ .
Sanity check with $x^{2} + y^{2} = 1$ : $(- \frac{1}{2})^{2} + (\frac{3}{2})^{2} = \frac{1}{4} + \frac{3}{4} = 1$ . ✓ The point really is on the unit circle.

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Common Mistakes

Radians vs degrees: Calculators and most advanced math use radians by default. sin(30) in radians is not 0.5; always check mode.
Reference angle signs: Forgetting to apply the correct sign for sine/cosine/tangent in each quadrant leads to wrong signs.
Unit circle memorization gaps: Not knowing the 30-45-60 exact values forces calculator use and loses exact answers on exams.
Forgetting undef at π/2 +kπ for tan/sec/csc/cot: Those points have cos or sin zero.
θ in degrees on unit circle table: The table is for radians; convert if needed.
Negative angles: Equivalent to clockwise; use even/odd or add 2π k to positive.

Exercises for §4.5 →

4.6Trigonometric identity

Identities are the algebra rules for trig. They let us rewrite expressions to simplify, solve, or integrate.

Basic Trigonometric Identities

The identities below are the ones everything else is built from, so it is worth seeing the family side by side.

Theorem 4.17 (Pythagorean identities). For every angle $θ$ at which the functions involved are defined,

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

The second and third are the first one divided through by $cos^{2} θ$ and by $sin^{2} θ$ , so there is really only one fact here, wearing three costumes.

Proposition 4.18 (Reciprocal, quotient and parity identities). The remaining fundamental identities are

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},

together with parity: $cos$ is even, while $sin$ and $tan$ are odd.

These allow simplification and are the starting point for proving more complex identities.

Intuition. Identities are like the " $a^{2} - b^{2} = (a - b) (a + b)$ " rules of algebra, but for trig: swap-in replacements that are true for every angle. The Pythagorean identity is the workhorse — it's just the equation of the unit circle ( $x^{2} + y^{2} = 1$ ) written with $x = cos θ$ and $y = sin θ$ . So if you know one of sine/cosine, this one fact hands you the other (up to sign).

Example 4.19 (Prove that $\frac{s i n θ}{t a n θ} = cos θ$ (wherever $tan θ$ is defined and $sin θ \neq = 0$ )). We start from the more complicated left side and transform it into the right side using known identities.

Solution. Rewrite $tan θ$ with the quotient identity: $\frac{sin θ}{tan θ} = \frac{sin θ}{sin θ / cos θ}$ . Dividing by a fraction is multiplying by its reciprocal: $= sin θ \cdot \frac{cos θ}{sin θ}$ . Cancel $sin θ$ (valid since $sin θ \neq = 0$ ): $= cos θ$ . The left side now equals the right side, so the identity holds on its domain. ✓□

Example 4.20 (Given $sin θ = \frac{3}{5}$ with $θ$ in quadrant I, find $cos θ$ and $tan θ$ ). Use the Pythagorean identity $sin^{2} θ + cos^{2} θ = 1$ : $(\frac{3}{5})^{2} + cos^{2} θ = 1$ , so $\frac{9}{25} + cos^{2} θ = 1$ .

Solution.

Isolate: $cos^{2} θ = 1 - \frac{9}{25} = \frac{16}{25}$ .
Take the square root and choose the sign from the quadrant. In quadrant I cosine is positive, so $cos θ = + \frac{4}{5}$ (this sign choice is the step students most often skip).
Then $tan θ = \frac{sin θ}{cos θ} = \frac{3/5}{4/5} = \frac{3}{4}$ .
Sanity check: $(\frac{3}{5})^{2} + (\frac{4}{5})^{2} = \frac{9}{25} + \frac{16}{25} = 1$ . ✓

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Common Mistakes

Identity direction: sin² + cos² =1 can be used to find one from the other, but you must take ± square root and choose sign from quadrant info.
Forgetting to state domain restrictions: Identities hold where the expressions are defined (e.g. tan defined).
Using = when proving: Start with one side, transform with = steps to other side; don't assume the thing to prove in middle.
Confusing identities with equations to solve: Identities are true for all (in domain); equations are true for specific values.
Sign errors in even/odd: cos even, sin tan odd.
Quotient vs reciprocal: tan = sin/cos (quotient), csc=1/sin (reciprocal).

Exercises for §4.6 →

About the practice questions. They check that you can carry out this chapter's computations correctly, and each one is graded on a single answer. They are not proof exercises: working through them confirms the mechanics, not that you could prove the results yourself. For that, re-read the statements above and try to reconstruct their proofs with the page closed.

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