Advanced Topics · Chapter 43

Geometric Trigonometry

Three formulas turn any triangle into numbers: half ab sin C for area, the Law of Sines with its circumradius 2R, and the Law of Cosines, the Pythagorean Theorem with a correction term.

What's this chapter about?

43.1 Area of a Triangle Using Trigonometry
43.2 Law of Sines
43.3 Law of Cosines
43.4 Practice Problems

You'll need

open on YouTube

43.10%

Area of a Triangle Using Trigonometry

In this section you will be able to find the area of any triangle from two sides and the angle between them: half of a times b times sin C.

Cut three corners off a 15-20-25 right triangle. Each corner has two sides of 5 and an angle you already know. That’s all you need for its area.

You know area as half base times height. Trig hands you the height: if sides $a$ and $b$ meet at the angle $C$ , the height onto $a$ is $b \sin C$ . So the area is $\frac12 ab \sin C$ , and you never have to drop the height yourself.

Look:

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

. Two different triangles with the same two sides can have exactly the same area.

Exploreplay first, formulas later

Try this: Keep a = 8 and b = 9 and step the angle C. For which C is the area largest? Why does sin C explain it?
Try this: Press "Use 180° − C". The third side changes, but the area doesn't. Can you find two such triangles whose third sides are whole numbers?
Try this: Drag A straight sideways. The height h stays put. What happens to the area?

Name it

Theorem 43.1.1 · Area of a Triangle Using Trigonometry

In a triangle with side lengths $a$ , $b$ , $c$ , where the angle between sides $a$ and $b$ is denoted by $C$ ,

\text{Area of the triangle} = \frac{1}{2} \cdot ab \cdot \sin C.

Watch itpredict, then reveal

Example 43.1

AIME

In $\triangle PQR$ , $PR = 15$ , $QR = 20$ , and $PQ = 25$ . Points $A$ and $B$ lie on $\overline{PQ}$ , points $C$ and $D$ lie on $\overline{QR}$ , and points $E$ and $F$ lie on $\overline{PR}$ , with $PA = QB = QC = RD = RE = PF = 5$ . Find the area of hexagon $ABCDEF$ .

1
Compare $15^2 + 20^2$ with $25^2$ . What kind of triangle is $PQR$ ?

open on YouTube

Try it

Problem 43.4.4

AMC 12

This problem also appears in the book as 35.6.

Regular octagon $ABCDEFGH$ has area $n$ . Let $m$ be the area of quadrilateral $ACEG$ . What is $\frac{m}{n}$ ?

Problem 43.4.5

AMC 12

Two non-congruent triangles have the same area. Each triangle has sides of length $8$ and $9$ , and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?

Contest cornerreal contest problems

Problem 43.4.1

AMC 12

This problem also appears in the book as 31.3.1.

Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3 \cdot AB$ . Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3 \cdot BC$ , and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3 \cdot CA$ . What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ ?

Problem 43.4.14(additional)

AIME

Hexagon $ABCDEF$ is divided into five rhombuses, $\mathcal{P}$ , $\mathcal{Q}$ , $\mathcal{R}$ , $\mathcal{S}$ , and $\mathcal{T}$ , as shown. Rhombuses $\mathcal{P}$ , $\mathcal{Q}$ , $\mathcal{R}$ , and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}$ . Let $K$ be the area of rhombus $\mathcal{T}$ . Given that $K$ is a positive integer, find the number of possible values for $K$ .

Checkpoint for section 43.1

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

43.20%

Law of Sines

In this section you will be able to use the Law of Sines to trade sides for angles, and spot the circumradius hiding in every side-angle pair.

Every triangle fits in exactly one circle. Its diameter is hiding in each side: divide a side by the sine of the opposite angle and you get it.

The Law of Sines says each side is proportional to the sine of the angle opposite it. Know two angles and one side, and you know the whole triangle.

Look:Fixed angles mean fixed sines, so the sides come out in the ratio

\sin A : \sin B : \sin C

Exploreplay first, formulas later

Try this: Drag A along the circumcircle. The angle A never changes. Why does a / sin A stay equal to the diameter 2R?
Try this: Look at the right triangle B C B′ on the diameter. Which angle in it equals A?
Try this: Press "SSA case" and switch the givens to SSA. Can one angle and two sides give two different triangles?

Name it

Theorem 43.2.1 · Law of Sines

In a triangle with sides $a$ , $b$ , $c$ and angles $A$ , $B$ , $C$ , where side $a$ is opposite angle $A$ , side $b$ is opposite angle $B$ , and side $c$ is opposite angle $C$ ,

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,

where $R$ is the circumradius of the triangle.

Try it

Problem 43.4.13(additional)

AMC 12

A circle centered at $O$ has radius $1$ and contains the point $A$ . The segment $\overline{AB}$ is tangent to the circle at $A$ and $\angle AOB = \theta$ . If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$ , then $OC = ?$ (Answer in terms of $\theta$ ; type θ, theta or t.)

Contest cornerreal contest problems

Problem 43.4.20(additional)

AMC 12

In $\triangle ABC$ with integer side lengths, $\cos A = \frac{11}{16}$ , $\cos B = \frac{7}{8}$ , and $\cos C = -\frac{1}{4}$ . What is the least possible perimeter for $\triangle ABC$ ?

Problem 43.4.19(additional)

AMC 12

In $\triangle PAT$ , $\angle P = 36^\circ$ , $\angle A = 56^\circ$ , and $PA = 10$ . Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA}$ , respectively, so that $PU = AG = 1$ . Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG}$ , respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA$ ?

Checkpoint for section 43.2

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

In a triangle, side $a = 10$ is opposite the angle $A = 30^\circ$ . What is the circumradius $R$ ?

In $\triangle ABC$ , $A = 45^\circ$ , $B = 60^\circ$ and $a = 2\sqrt2$ . How long is side $b$ ?

In $\triangle ABC$ , $B = 30^\circ$ , $a = BC = 10$ and $b = CA = 6$ . Use the Law of Sines: what is $\sin A$ ? (The figure shows why two different triangles fit.)

43.30%

Law of Cosines

In this section you will be able to use the Law of Cosines, the Pythagorean Theorem with a correction term, to get a side from an angle or an angle from three sides.

The Pythagorean Theorem needs a right angle. Bend the angle and $c^2$ drifts away from $a^2 + b^2$ . The Law of Cosines tells you exactly how far.

When $C$ is acute, $c^2$ is short of $a^2 + b^2$ by $2ab\cos C$ . When $C$ is obtuse, $\cos C < 0$ and $c^2$ is bigger. Run it backwards and three sides give you any angle.

Look:

120^\circ

angles are everywhere in contest problems. There

\cos C = -\frac12

, so

c^2 = a^2 + b^2 + ab

Exploreplay first, formulas later

Try this: Set C = 90°. What happens to the correction 2ab cos C? Which theorem is left?
Try this: Step C up to 120°. cos 120° = −1/2, so the correction turns into a surplus of exactly ab. Check the bars.
Try this: Switch the givens to SSS. Which angle is largest? Is it always the one opposite the longest side?

Name it

Theorem 43.3.1 · Law of Cosines

In a triangle with side lengths $a$ , $b$ , $c$ , where the angle between sides $a$ and $b$ is denoted by $C$ ,

c^2 = a^2 + b^2 - 2ab \cdot \cos C.

Watch itpredict, then reveal

Example 43.2

AIME

Triangle $ABC$ has right angle at $B$ , and contains a point $P$ for which $PA = 10$ , $PB = 6$ , and $\angle APB = \angle BPC = \angle CPA$ . Find $PC$ .

1
The three equal angles at $P$ go all the way around. How big is each?

Predict first:
Your prediction

open on YouTube

Example 43.3

AMC 12

Circles $\omega_1$ , $\omega_2$ , and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$ , $P_2$ , and $P_3$ lie on $\omega_1$ , $\omega_2$ , and $\omega_3$ respectively such that $P_1P_2 = P_2P_3 = P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i = 1, 2, 3$ , where $P_4 = P_1$ . See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a} + \sqrt{b}$ for positive integers $a$ and $b$ . What is $a + b$ ?

1
Let $O_1, O_2, O_3$ be the centers. Tangent circles of radius $4$ : how far apart are two centers?

Predict first:
Your prediction

open on YouTube

Try it

Problem 43.4.8(additional)

AHSME

In $\triangle ABC$ , side $a = \sqrt{3}$ , side $b = \sqrt{3}$ , and side $c > 3$ . Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$ . Then $x$ equals:

Problem 43.4.10(additional)

AMC 10

This problem also appears in the book as 31.3.4.

In rectangle $ABCD$ , $\overline{AB} = 20$ and $\overline{BC} = 10$ . Let $E$ be a point on $\overline{CD}$ such that $\angle CBE = 15^\circ$ . What is $\overline{AE}$ ?

Contest cornerreal contest problems

Problem 43.4.2

AMC 10/12

Triangle $\triangle ABC$ has side lengths $AB = 80$ , $BC = 45$ , and $AC = 75$ . The bisector of $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P$ . What is $BP$ ?

Problem 43.4.9(additional)

AHSME

In triangle $ABC$ , $3\sin A + 4\cos B = 6$ and $4\sin B + 3\cos A = 1$ . Then $\angle C$ in degrees is

Problem 43.4.15(additional)

AMC 12

Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

Problem 43.4.16(additional)

AIME

In triangle $ABC$ , $\tan \angle CAB = 22/7$ , and the altitude from $A$ divides $BC$ into segments of length $3$ and $17$ . What is the area of triangle $ABC$ ?

Checkpoint for section 43.3

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

43.4

Practice Problems

In this section you will be able to mix the area formula, the Law of Sines and the Law of Cosines on contest problems with polygons, circles and rotations.

Regular polygons, rings of circles, folded paper and a rotated line: each one hides a triangle you can now solve.

Look for the triangle first: two sides and the angle between them, or three sides, or a side and two angles. Then pick the formula that fits.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 43.4.3

AMC 12

An analog clock starts at midnight and runs for $2025$ minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?

Problem 43.4.6

AMC 12

A circle of radius $r$ is surrounded by $12$ circles of radius $1$ , externally tangent to the central circle and sequentially tangent to each other, as shown. Then $r$ can be written as $\sqrt{a} + \sqrt{b} + c$ , where $a, b, c$ are integers. What is $a + b + c$ ?

Problem 43.4.7

AMC 10/12

A regular pentagon with area $\sqrt{5} + 1$ is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

Problem 43.4.11(additional)

AMC 12

In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$ . The enclosed area of the hexagon is $6\sqrt{3}$ . What is the perimeter of the hexagon?

Problem 43.4.12(additional)

AIME

Let $ABCDEF$ be a regular hexagon. Let $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , and $AF$ , respectively. The segments $\overline{AH}$ , $\overline{BI}$ , $\overline{CJ}$ , $\overline{DK}$ , $\overline{EL}$ , and $\overline{FG}$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ABCDEF$ be expressed as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 43.4.17(additional)

AIME

This problem also appears in the book as 33.6.29.

In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$ , side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$ , $MN = 65$ , and $KL = 28$ . The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$ . Find $MO$ .

Problem 43.4.18(additional)

AIME

This problem also appears in the book as 43.1.

Problem 43.4.21(additional)

AMC 12

Line $\ell$ in the coordinate plane has equation $3x - 5y + 40 = 0$ . This line is rotated $45^\circ$ counterclockwise about the point $(20, 20)$ to obtain line $k$ . What is the $x$ -coordinate of the $x$ -intercept of line $k$ ?

Problem 43.4.22(additional)

AIMEChallenge

Let $\triangle ABC$ have side lengths $AB = 30$ , $BC = 32$ , and $AC = 34$ . Point $X$ lies in the interior of $\overline{BC}$ , and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$ , respectively. Find the minimum possible area of $\triangle AI_1I_2$ as $X$ varies along $\overline{BC}$ .