42. Algebraic Trigonometry · Mastering AMC 10/12

42.10%

Six Trig Ratios

In this section you will be able to read sine, cosine, tangent and their three reciprocals off a right triangle.

A 10 m ladder leans on a wall at 60° to the ground. How high does it reach? Opposite over hypotenuse: the height is $10\sin 60^\circ$ .

Every right triangle with the same acute angle $a$ has the same shape, so the ratios of its sides depend only on $a$ . Those ratios get names: sine, cosine and tangent, plus their flips cosecant, secant and cotangent.

This chapter is mainly for the AMC 12, but the same tools make some AMC 10 problems much shorter.

Exploreplay first, formulas later

Angle a40°

a = 40^\circ

\sin a=\tfrac{\text{opp}}{\text{hyp}},\ \cos a=\tfrac{\text{adj}}{\text{hyp}},\ \tan a=\tfrac{\text{opp}}{\text{adj}}

f	f(40°)
sin a	$0.6428$
cos a	$0.766$
tan a	$0.8391$
csc a	$1.5557$
sec a	$1.3054$
cot a	$1.1918$

Try this: Drag the angle to 30°. Which ratio is exactly 1/2? Now try 60°: which ratio is 1/2 there?
Try this: The hypotenuse here is 1, so the opposite side is sin a itself. What happens to the triangle as a approaches 90°?
Try this: Why is csc a = 1/sin a? Compare the csc and sin rows at 30° and at 45°.

Name it

Theorem 42.1.1 · Trigonometric Identities

In a right triangle with an acute angle $a$ :

\text{Sine: } \sin(a) = \frac{\text{Opposite}}{\text{Hypotenuse}}

\text{Cosine: } \cos(a) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

\text{Tangent: } \tan(a) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(a)}{\cos(a)}

Theorem 42.2.1 · More Trigonometric Identities

\text{Cosecant: } \csc(a) = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{1}{\sin(a)}

\text{Secant: } \sec(a) = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{1}{\cos(a)}

\text{Cotangent: } \cot(a) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{\tan(a)} = \frac{\cos(a)}{\sin(a)}

Checkpoint for section 42.1

Get 2 of 3 right to master this section. One try each, no hints.

42.30%

Special Angles and the Unit Circle

In this section you will be able to know the exact values at special angles and use the unit circle to handle any angle.

What is $\cos 150^\circ$ ? No right triangle has a $150^\circ$ angle. Put the angle on a circle of radius 1 instead: its cosine is just the $x$ -coordinate of the point.

On the unit circle, the angle $a$ ends at the point $P = (\cos a, \sin a)$ . That works for every angle, not only acute ones. The $30$ - $60$ - $90$ and $45$ - $45$ - $90$ triangles give the exact values below; the reflections in Theorem 42.4.1 carry them to the other quadrants.

\begin{array}{c|c|c} a & \cos a & \sin a \ \hline 0^\circ & 1 & 0 \[2pt] 15^\circ & \frac{\sqrt6+\sqrt2}{4} & \frac{\sqrt6-\sqrt2}{4} \[2pt] 30^\circ & \frac{\sqrt3}{2} & \frac12 \[2pt] 45^\circ & \frac{\sqrt2}{2} & \frac{\sqrt2}{2} \[2pt] 60^\circ & \frac12 & \frac{\sqrt3}{2} \[2pt] 75^\circ & \frac{\sqrt6-\sqrt2}{4} & \frac{\sqrt6+\sqrt2}{4} \[2pt] 90^\circ & 0 & 1 \[2pt] 120^\circ & -\frac12 & \frac{\sqrt3}{2} \[2pt] 135^\circ & -\frac{\sqrt2}{2} & \frac{\sqrt2}{2} \[2pt] 150^\circ & -\frac{\sqrt3}{2} & \frac12 \[2pt] 180^\circ & -1 & 0 \end{array}

The book writes $\cos 15^\circ = \frac{\sqrt3+1}{2\sqrt2}$ ; multiply top and bottom by $\sqrt2$ to get $\frac{\sqrt6+\sqrt2}{4}$ . Look: the book’s table also repeats the $0^\circ$ row, prints $\cos 180^\circ = 1$ (it is $-1$ ) and writes $\sin 0^\circ$ where it means $\sin 180^\circ$ . The table above is fixed.

Exploreplay first, formulas later

Angle a30°

a = 30^\circ

f	f(30°)
cos a	$\frac{\sqrt3}{2}$
sin a	$\frac12$
tan a	$\frac{\sqrt3}{3}$

sin²a + cos²a = 0.25 + 0.75 = 1

Reflect a to

Try this: Press 180° − a with a = 30°. The point moves to 150°. What happens to cos? To sin?
Try this: Drag the angle anywhere. The bar shows sin² a + cos² a. Does it ever stop adding up to 1?
Try this: Turn on Radians. 30° is π/6. What is 135° in radians?

Name it

Theorem 42.4.1 · Unit Circle Identities

\sin(-a) = -\sin(a)

\sin(a) = \sin(180^\circ - a)

\cos(a) = \cos(-a)

\cos(a) = -\cos(180^\circ - a)

\tan(a) = -\tan(180^\circ - a)

\tan(-a) = -\tan(a)

Theorem 42.5.1 · Pythagorean Identities

\sin^2(a) + \cos^2(a) = 1

\tan^2(a) + 1 = \sec^2(a)

\cot^2(a) + 1 = \csc^2(a)

Angle a40°

a = 40^\circ

P=(\cos a,\ \sin a)

f	f(40°)
cos a	$0.766$
sin a	$0.6428$
tan a	$0.8391$

Try it

Problem 42.13.9(additional)

AMC 12

The functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$ . What is the least period of the function $\cos(\sin(x))$ ?

Checkpoint for section 42.3

Get 2 of 3 right to master this section. One try each, no hints.

1

Use the $30$ - $60$ - $90$ triangle: what is $\sin60^\circ$ ?

short leg a1

Drag the purple dot or use the stepper to resize. The ratios never change.

Area =

\frac{\sqrt{3}}{2}

≈ 0.87

a : a\sqrt3 : 2a,\quad \text{Area} = \tfrac{\sqrt3}{2}a^2

Equilateral: $\text{Area} = \tfrac{\sqrt3}{4}a^2$
45-45-90: $\text{Area} = \tfrac12 a^2$
30-60-90: $\text{Area} = \tfrac{\sqrt3}{2}a^2$

2

Use the $45$ - $45$ - $90$ triangle and Theorem 42.4.1: what is $\cos135^\circ$ ?

3

Angle $a$ is obtuse and $\sin a = \frac35$ . What is $\cos a$ ?

42.60%

Double, Sum and Half Angles

In this section you will be able to expand sin and cos of a sum, a difference, a double angle and a half angle.

You know $\cos 45^\circ$ and $\cos 30^\circ$ . What about $\cos 15^\circ$ ? Since $15^\circ = 45^\circ - 30^\circ$ , one formula turns the values you know into the one you want.

Rotating by $a$ and then by $b$ is the same as rotating by $a + b$ . Writing that down in coordinates gives the addition formulas. Everything else in this section follows from them: set $b = a$ for the double angle, then solve for the half angle.

Exploreplay first, formulas later

a45°

b30°

\sin(45^\circ+30^\circ) = 0.9659

= \frac{\sqrt2}{2}\cdot \frac{\sqrt3}{2} + \frac12\cdot \frac{\sqrt2}{2} = 0.9659

\sin(a+b) = \sin a\cos b + \sin b\cos a

✓ Both sides agree

Try this: Rotate by a, then by b: the third arrow lands at a + b. Its coordinates are cos(a + b) and sin(a + b). Check both sides of the formula.
Try this: Pick Difference and cos(a−b) with a = 45°, b = 30°. You just computed cos 15° exactly.
Try this: Pick Double. Why is the double-angle formula just the sum formula with b = a?

Name it

Theorem 42.6.1 · Double Angle Identities

\sin(2a) = 2\sin(a)\cos(a)

\cos(2a) = \cos^2(a) - \sin^2(a) = 2\cos^2(a) - 1 = 1 - 2\sin^2(a)

\tan(2a) = \frac{2\tan(a)}{1 - \tan^2(a)}

Theorem 42.7.1 · Addition and Subtraction Identities

\sin(a + b) = \sin(a)\cos(b) + \sin(b)\cos(a)

\sin(a - b) = \sin(a)\cos(b) - \sin(b)\cos(a)

\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) = \frac{\tan(a) + \tan(b)}{1 - \tan a \tan b}

\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan a \tan b}

Theorem 42.8.1 · Half Angle Identities

\sin\left(\frac a2\right) = \pm\sqrt{\frac{1 - \cos(a)}{2}}

\cos\left(\frac a2\right) = \pm\sqrt{\frac{1 + \cos(a)}{2}}

The sign is the sign of $\sin\frac a2$ (or $\cos\frac a2$ ): look at the quadrant of $\frac a2$ .

Try it

Problem 42.13.13(additional)

AMC 12

If $\sum_{n=0}^{\infty}\cos^{2n}\theta = 5$ , what is the value of $\cos2\theta$ ?

Problem 42.13.10(additional)

AMC 12

The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac{1}{\sqrt3}$ . For $n \ge 1$ ,

a_{n+2} = \frac{a_n + a_{n+1}}{1 - a_na_{n+1}}.

What is $|a_{2009}|$ ?

Contest cornerreal contest problems

Problem 42.13.17(additional)

AIME

Given that $(1 + \sin t)(1 + \cos t) = \frac54$ and

(1 - \sin t)(1 - \cos t) = \frac mn - \sqrt k,

where $k$ , $m$ , and $n$ are positive integers with $m$ and $n$ relatively prime, find $k + m + n$ .

Checkpoint for section 42.6

Get 2 of 3 right to master this section. One try each, no hints.

42.90%

Sum to Product and Back

In this section you will be able to switch between sums and products of sines and cosines.

$\sin a + \sin b = \sqrt{5/3}$ and $\cos a + \cos b = 1$ look like two equations in two unknowns. They are really the coordinates of one vector: the sum of two unit vectors.

Add the unit vectors at angles $a$ and $b$ . By symmetry the sum points at the average angle $\frac{a+b}{2}$ , and its length is $2\cos\frac{a-b}{2}$ . Read off its coordinates and you have the sum-to-product formulas. Run them backwards and products become sums.

Exploreplay first, formulas later

a80°

b20°

\sin 80^\circ + \sin 20^\circ = 1.3268

= 2\sin 50^\circ\cos 30^\circ = 1.3268

\sin a+\sin b = 2\sin\frac{a+b}{2}\cos\frac{a-b}{2}

✓ Both sides agree

length of the sum = 1.7321

Try this: Two unit vectors at a and b, tip to tail. How long is their sum? Compare it with 2cos((a − b)/2).
Try this: The sum vector points at the average angle (a + b)/2. Square and add its coordinates: which identity shows up?
Try this: Pick Product → sum and cos a · cos b. Why is the x-coordinate of M equal to cos a cos b?

Name it

Theorem 42.9.1 · Sum to Product Identities

\sin(a) + \sin(b) = 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)

\sin(a) - \sin(b) = 2\sin\left(\frac{a-b}{2}\right)\cos\left(\frac{a+b}{2}\right)

\cos(a) + \cos(b) = 2\cos\left(\frac{a-b}{2}\right)\cos\left(\frac{a+b}{2}\right)

\cos(a) - \cos(b) = -2\sin\left(\frac{a-b}{2}\right)\sin\left(\frac{a+b}{2}\right)

Theorem 42.10.1 · Product to Sum Identities

\sin(a)\sin(b) = \frac12\left(\cos(a - b) - \cos(a + b)\right)

\cos(a)\cos(b) = \frac12\left(\cos(a - b) + \cos(a + b)\right)

\sin(a)\cos(b) = \frac12\left(\sin(a + b) + \sin(a - b)\right)

Watch itpredict, then reveal

Example 42.1

AMC 12

Suppose that $\sin a + \sin b = \sqrt{\frac53}$ and $\cos a + \cos b = 1$ . What is $\cos(a - b)$ ?

1
Think of $(\cos a, \sin a)$ and $(\cos b, \sin b)$ as two unit vectors. Placed tip to tail, they end at $(\cos a + \cos b,\ \sin a + \sin b) = \left(1, \sqrt{\frac53}\right)$ . What is the squared length of that sum vector?

Predict first:
Your prediction

open on YouTube

Example 42.2

AIME

Let $x$ be a real number such that $\sin^{10}x + \cos^{10}x = \frac{11}{36}$ . Then $\sin^{12}x + \cos^{12}x = \frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

1
Let $s = \sin^2 x$ and $c = \cos^2 x$ , so $s + c = 1$ . Call $p = sc$ . Write $s^2 + c^2$ in terms of $p$ .

Predict first:
Your prediction

open on YouTube

Contest cornerreal contest problems

Problem 42.13.16(additional)

AIME

Given that $A_k = \frac{k(k-1)}{2}\cos\frac{k(k-1)\pi}{2}$ , find $|A_{19} + A_{20} + \cdots + A_{98}|$ .

Checkpoint for section 42.9

Get 2 of 3 right to master this section. One try each, no hints.

42.110%

Graphs, Periods and Counting Solutions

In this section you will be able to use periods and graphs to count the solutions of a trig equation.

How many solutions does $\sin x = \sin 5x$ have on $[0, \pi]$ ? Solving is messy. Drawing both graphs and counting the crossings is fast.

A graph shows every solution at once. Watch for three traps: a curve that only touches the other one (a tangency counts once), a solution sitting exactly on an endpoint (it counts only if the interval is closed), and the asymptotes of $\tan$ , where the graph jumps without crossing anything.

Exploreplay first, formulas later

Try this: For n = 2, 3, 4, 5 count the solutions of sin x = sin nx on [0, π]. For which n does a tangency (a ring) appear?
Try this: The square marks are solutions at a closed end of the interval. What would change on the open interval (0, π)?
Try this: Why does tan repeat after π, not 2π? Press Show me on the definition card.

Name it

Definition 42.11.1 · Periodic function

A periodic function is a function that repeats a pattern of $y$ -values at regular intervals. One complete repetition of the pattern is called a cycle. The period of a function is the horizontal length of one complete cycle.

The period of $\sin$ and $\cos$ is $2\pi$ ; the period of $\tan$ is $\pi$ .

Fixed from the book. The book says the period of $\tan$ is $2\pi$ . It repeats after only $\pi$ , because $\tan(x + \pi) = \frac{-\sin x}{-\cos x} = \tan x$ . (The book also calls every periodic function "trigonometric"; periodic is the general word.)

Concept 42.12.1 · Sine Graph

The graph of $y = \sin x$ is a wave between $-1$ and $1$ . It starts at $(0, 0)$ , peaks at $x = \frac\pi2$ , crosses zero at every multiple of $\pi$ , and repeats every $2\pi$ .

Concept 42.12.2 · Cosine Graph

The graph of $y = \cos x$ is the sine wave shifted left by $\frac\pi2$ : $\cos x = \sin\left(x + \frac\pi2\right)$ . It starts at its peak $(0, 1)$ , crosses zero at odd multiples of $\frac\pi2$ , and repeats every $2\pi$ .

Concept 42.12.3 · Tan Graph

The graph of $y = \tan x$ crosses zero at every multiple of $\pi$ and shoots off to $\pm\infty$ at the vertical asymptotes $x = \frac\pi2 + k\pi$ , where $\cos x = 0$ . It repeats every $\pi$ .

Remark 42.12.4

Long trigonometric expressions can be evaluated by telescoping, by using identities in clever ways, or by complex number substitutions (see Chapter 44, Complex Numbers).

Watch itpredict, then reveal

Example 42.3

AMC 12

For each integer $n > 1$ , let $F(n)$ be the number of solutions to the equation $\sin x = \sin(nx)$ on the interval $[0, \pi]$ . What is $\displaystyle\sum_{n=2}^{2007} F(n)$ ?

1
Start small: $n = 2$ . Sketch $y = \sin x$ and $y = \sin 2x$ on $[0, \pi]$ . How many times do they meet? Count the endpoints too.

Predict first:
Your prediction

open on YouTube

Try it

Problem 42.13.1

AMC 12

How many solutions does the equation $\sin\left(\frac\pi2\cos x\right) = \cos\left(\frac\pi2\sin x\right)$ have in the closed interval $[0, \pi]$ ?

Contest cornerreal contest problems

Problem 42.13.6

AMC 12

How many values of $\theta$ in the interval $0 < \theta \le 2\pi$ satisfy

1 - 3\sin\theta + 5\cos3\theta = 0?

Problem 42.13.14(additional)

AMC 12

How many solutions does the equation $\tan(2x) = \cos\left(\frac x2\right)$ have on the interval $[0, 2\pi]$ ?

Problem 42.13.7

AMC 12

How many real numbers satisfy the equation $\sin(20\pi x) = \log_{20}(x)$ ?

Problem 42.13.12(additional)

AMC 12

The number of $x$ -intercepts on the graph of $y = \sin(1/x)$ in the interval $(0.0001, 0.001)$ is closest to

Checkpoint for section 42.11

Get 2 of 3 right to master this section. One try each, no hints.

42.13

Practice Problems

In this section you will be able to mix every tool of the chapter on contest problems.

The rest of the practice set, from a quick special-angle area to an AIME geometry problem. Decide first which tool each one needs.

Contest cornerreal contest problems

Problem 42.13.3

AMC 12

The three side lengths of a triangle are in arithmetic progression with shortest side of length 6. One interior angle measures $120^\circ$ . What is the area of the triangle?

Problem 42.13.11(additional)

AMC 12

An object moves 8 cm in a straight line from $A$ to $B$ , turns at an angle $\alpha$ , measured in radians and chosen at random from the interval $(0, \pi)$ , and moves 5 cm in a straight line to $C$ . What is the probability that $AC < 7$ ?

Problem 42.13.5

AMC 12

Usain is walking for exercise by zigzagging across a 100-meter by 30-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$ . He wants to increase the distance walked by zigzagging as shown in the figure below ( $APQRS$ ). What angle $\theta = \angle PAB = \angle QPC = \angle RQB = \cdots$ will produce a length that is 120 meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.)

Problem 42.13.15(additional)

AMC 12

Which of the following describes the largest subset of values of $y$ within the closed interval $[0, \pi]$ for which

\sin(x + y) \le \sin(x) + \sin(y)

for every $x$ between $0$ and $\pi$ , inclusive?

Problem 42.13.2

AMC 12

Suppose that the roots of the polynomial $P(x) = x^3 + ax^2 + bx + c$ are $\cos\frac{2\pi}{7}$ , $\cos\frac{4\pi}{7}$ , and $\cos\frac{6\pi}{7}$ , where angles are in radians. What is $abc$ ?

Problem 42.13.4

AMC 12

Cyrus the frog sits on a flat surface. He jumps, landing 2 feet away. He then chooses a direction at random and again jumps 2 feet. What is the probability that after the second jump Cyrus lands within 1 foot of his starting position?

Problem 42.13.18(additional)

AMC 12

Suppose $a$ is a real number such that the equation

a\cdot(\sin x + \sin(2x)) = \sin(3x)

has more than one solution in the interval $(0, \pi)$ . The set of all such $a$ can be written in the form $(p, q) \cup (q, r)$ , where $p$ , $q$ , and $r$ are real numbers with $p < q < r$ . What is $p + q + r$ ?

Problem 42.13.8

2024 AIME I Problem 10Challenge

Let $ABC$ be a triangle inscribed in circle $\omega$ . Let the tangents to $\omega$ at $B$ and $C$ intersect at point $P$ , and let $\overline{AP}$ intersect $\omega$ at $D$ . Find $AD$ , if $AB = 5$ , $BC = 9$ , and $AC = 10$ . (Write $AD = \frac mn$ in lowest terms and enter $m + n$ .)