VertexMath · MCR3U · Free worksheet

Trigonometric Ratios Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 50 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Right triangles: opposite and adjacent depend on which angle you stand at. The hypotenuse is always across from the $90^\circ$ angle.
CAST: All ratios are positive in Quadrant I, Sine in II, Tangent in III, Cosine in IV. A reference angle is always measured to the $x$ -axis.
Identities: $\sin^2\theta+\cos^2\theta=1$ , $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ , $\csc\theta=\dfrac{1}{\sin\theta}$ , $\sec\theta=\dfrac{1}{\cos\theta}$ , $\cot\theta=\dfrac{1}{\tan\theta}$ .
Sine law $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$ for AAS, ASA, SSA. Cosine law $c^2=a^2+b^2-2ab\cos C$ for SAS, SSS.
$\sin^{-1}$ only returns an acute angle, but $\sin\theta=\sin(180^\circ-\theta)$ . Always check whether the obtuse angle works too.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Right Triangles · Any Angle · Identities · Sine Law · Cosine Law

K1 Right-Triangle Ratios

Parts (a) and (b) use right triangle $PQR$ below, with $\angle Q=90^\circ$ .

a)
State $\sin P$ , $\cos P$ and $\tan P$ as fractions.
b)
Determine $\angle P$ to the nearest tenth of a degree.
c)
In a different right triangle $ABC$ , $\angle B=90^\circ$ , $\angle A=35^\circ$ and $BC=8$ cm. Determine $AC$ to one decimal place.

Stuck? Peek at a hint

Stand at vertex $P$ : the opposite side is the one that does not touch $P$ .

K2 Any Angle

a)
Complete the table: state the reference angle of each angle and the sign of $\tan\theta$ .

$\theta$ $110^\circ$ $250^\circ$ $335^\circ$
$\text{reference angle}$
$\text{sign of }\tan\theta$
b)
Determine the exact values of $\sin240^\circ$ , $\cos240^\circ$ and $\tan240^\circ$ .
c)
Determine the exact value of $\sin^2120^\circ+\cos225^\circ\tan135^\circ$ .

$\theta$	$110^\circ$	$250^\circ$	$335^\circ$
$\text{reference angle}$
$\text{sign of }\tan\theta$

Stuck? Peek at a hint

Sketch each angle in standard position and measure from the terminal arm to the nearest part of the $x$ -axis.

K3 Identities

Given $\sin\theta=-\dfrac{8}{17}$ and $270^\circ<\theta<360^\circ$ .

a)
Determine the exact value of $\cos\theta$ .
b)
Determine the exact values of $\tan\theta$ and $\csc\theta$ .
c)
Simplify $\dfrac{(\sin\theta+\cos\theta)^2-1}{\cos^2\theta}$ , then evaluate it for this $\theta$ .

Stuck? Peek at a hint

Use $\cos^2\theta=1-\sin^2\theta$ , then choose the sign of $\cos\theta$ from the quadrant.

K4 Sine Law

In $\triangle DEF$ , $\angle D=52^\circ$ , $\angle E=71^\circ$ and $f=15$ cm. Give lengths to one decimal place.

a)
Determine $d$ .
b)
Determine $e$ .
c)
In $\triangle XYZ$ , $x=11$ cm, $y=8$ cm and $\angle X=70^\circ$ . Determine $\angle Y$ to the nearest tenth of a degree.

Stuck? Peek at a hint

Find $\angle F$ first, so that side $f$ has a partner angle.

K5 Cosine Law

Give lengths to one decimal place and angles to the nearest tenth of a degree.

a)
In $\triangle ABC$ , $b=7$ cm, $c=10$ cm and $\angle A=48^\circ$ . Determine $a$ .
b)
In $\triangle KLM$ , $k=5$ m, $l=8$ m and $m=11$ m. Determine $\angle M$ .
c)
In $\triangle STU$ , $s=6$ cm, $t=9$ cm and $\angle U=125^\circ$ . Determine $u$ .

Stuck? Peek at a hint

Evaluate $2ab\cos C$ as one number before subtracting it from $a^2+b^2$ .

Real World (A)

Ambiguous Case · Cosine Law · 3D Problems

A1 Signal on the Highway

A straight highway in Northern Ontario starts at a town $A$ . A cell tower $T$ is 12 km from $A$ , and segment $AT$ makes an angle of $32^\circ$ with the highway. The tower's signal reaches 9 km in every direction.

a)
A car is at a point $C$ on the highway, exactly 9 km from the tower. Use the sine law to find the two possible measures of $\angle ACT$ , to the nearest tenth of a degree.
b)
How far from $A$ does a car driving away from town gain signal, and how many kilometres does it then drive with signal?
c)
To cut costs, the provider installs an antenna that reaches only 6 km. Repeat the sine-law step from (a). What does the result mean for drivers on this highway?

Stuck? Peek at a hint

Sketch $\triangle ACT$ : the $32^\circ$ angle at $A$ is opposite the 9 km side.

A2 Flower Bed

A landscaper in Kitchener is building a triangular flower bed with edges 4.0 m, 5.5 m and 8.0 m long.

a)
Determine the largest angle of the bed, to the nearest tenth of a degree.
b)
A tree goes in the corner between the 4.0 m and 8.0 m edges. Determine the angle of that corner.
c)
A client asks for a second bed with edges 3.0 m, 4.5 m and 8.0 m. Use the cosine law to find its largest angle. What should the landscaper tell the client?

Stuck? Peek at a hint

The largest angle is opposite the longest edge; put that edge alone on the left side of the cosine law.

A3 Balloon Spotting

Two observers stand on level farmland at $P$ and $Q$ , with $Q$ 3.0 km due east of $P$ . A hot-air balloon hovers directly above point $B$ on the ground. From $P$ , $B$ is on a bearing of $050^\circ$ ; from $Q$ , $B$ is on a bearing of $300^\circ$ . From $P$ , the angle of elevation of the balloon is $20^\circ$ .

Top view of the ground

a)
Determine the three interior angles of $\triangle PQB$ on the ground.
b)
Determine the height of the balloon, to the nearest metre.
c)
The observer at $P$ has a laser rangefinder that measures straight-line distances up to 1.65 km. Can it measure the distance to the balloon? Explain.

Stuck? Peek at a hint

Bearings are measured clockwise from north; find the angle between $PB$ and $PQ$ , which points due east.

Think It Through (T)

Error Analysis · Explain Why

T1 Proof or Not?

A student wants to prove $\dfrac{\cos x}{1-\sin x}=\dfrac{1+\sin x}{\cos x}$ and writes:
"Cross-multiply: $\cos^2x=(1-\sin x)(1+\sin x)=1-\sin^2x$ . That is the Pythagorean identity, which is true, so the identity is proved."

a)
Explain why this is not a valid proof.
b)
Write a valid proof, working on the left side only.
c)
A second student claims $\sin x+\cos x=1$ is an identity because it works for $x=0^\circ$ and $x=90^\circ$ . Find a counterexample.

Stuck? Peek at a hint

For (b), multiply the left side by $\dfrac{1+\sin x}{1+\sin x}$ .

T2 The Hidden Angle

A student solves $\triangle ABC$ with $a=6$ , $b=9$ and $c=13$ . They use the cosine law to get $\angle A\approx23.9^\circ$ , then the sine law to get $\angle C\approx61.2^\circ$ , and finally $\angle B\approx180^\circ-23.9^\circ-61.2^\circ=94.9^\circ$ .

a)
Use the cosine law to find $\angle C$ . Which of the student's angles are wrong?
b)
Explain why the sine law gave $61.2^\circ$ .
c)
Keep $a=6$ and $b=9$ . Find all lengths $c$ for which $\angle C$ is obtuse.

Stuck? Peek at a hint

For (c), $\angle C$ is obtuse exactly when $\cos C<0$ in the cosine law.

Boss Level

Optional challenge

Tower Walk

Point $A$ is on level ground, due south of the base of a vertical tower of height $h$ metres. From $A$ , the angle of elevation of the top is $45^\circ$ . A hiker walks 100 m due east from $A$ to $B$ , where the angle of elevation of the top is $30^\circ$ .

a)
Write the distances from $A$ and from $B$ to the base of the tower in terms of $h$ , exactly.
b)
Find $h$ exactly.
c)
Find the angle of elevation of the top from $M$ , the midpoint of $AB$ , to the nearest tenth of a degree.

Stuck? Peek at a hint

Draw a top view first: mark the base of the tower, $A$ and $B$ , and the north and east directions.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $\sin P=\frac{15}{17}$ , $\cos P=\frac{8}{17}$ , $\tan P=\frac{15}{8}$
b) $\angle P\approx61.9^\circ$
c) $AC=\dfrac{8}{\sin35^\circ}\approx13.9$ cm

a) $110^\circ$ : $70^\circ$ , negative; $250^\circ$ : $70^\circ$ , positive; $335^\circ$ : $25^\circ$ , negative
b) $\sin240^\circ=-\frac{\sqrt3}{2}$ , $\cos240^\circ=-\frac12$ , $\tan240^\circ=\sqrt3$
c) $\dfrac{3+2\sqrt2}{4}$

a) $\cos\theta=\frac{15}{17}$
b) $\tan\theta=-\frac{8}{15}$ , $\csc\theta=-\frac{17}{8}$
c) $2\tan\theta$ ; $-\frac{16}{15}$

a) $d\approx14.1$ cm
b) $e\approx16.9$ cm
c) $\angle Y\approx43.1^\circ$

a) $a\approx7.4$ cm
b) $\angle M\approx113.6^\circ$
c) $u\approx13.4$ cm

Real World

a) $\angle ACT\approx45.0^\circ$ or $135.0^\circ$
b) It gains signal $3.8$ km from $A$ and loses it $16.5$ km from $A$ : $12.7$ km with signal.
c) $\sin\angle ACT=\frac{12\sin32^\circ}{6}\approx1.06>1$ : no such angle, so no point of the highway is within 6 km of the tower (the closest is about $6.4$ km). Drivers never get signal from it.

a) $113.8^\circ$ , at the corner opposite the 8.0 m edge
b) $39.0^\circ$
c) $\cos C=\frac{3^2+4.5^2-8^2}{2(3)(4.5)}\approx-1.29<-1$ : no such angle. The bed can't be built: $3.0+4.5<8.0$ , so the two short edges can't meet.

a) $\angle P=40^\circ$ , $\angle Q=30^\circ$ , $\angle B=110^\circ$
b) $PB\approx1.60$ km, so the height is about $581$ m
c) No. The rangefinder measures the line of sight, $\frac{PB}{\cos20^\circ}\approx1.70$ km, which is more than 1.65 km, even though the ground distance $PB\approx1.60$ km is within range.

Think It Through

a) Cross-multiplying assumes the two sides are already equal and changes both at once. A proof must start from one side and transform it into the other.
b) LS $=\frac{\cos x}{1-\sin x}\cdot\frac{1+\sin x}{1+\sin x}=\frac{\cos x(1+\sin x)}{1-\sin^2x}=\frac{\cos x(1+\sin x)}{\cos^2x}=\frac{1+\sin x}{\cos x}=$ RS
c) e.g. $x=45^\circ$ : $\sin45^\circ+\cos45^\circ=\sqrt2\approx1.41\ne1$

a) $\angle C\approx118.8^\circ$ . $\angle C$ and $\angle B$ are wrong ( $\angle B\approx37.4^\circ$ ); $\angle A$ is correct.
b) $\sin C\approx0.876$ has two solutions between $0^\circ$ and $180^\circ$ , $61.2^\circ$ and $118.8^\circ$ ; the calculator returns only the acute one. Finding the largest angle first with the cosine law avoids this.
c) $3\sqrt{13}<c<15$ (about $10.8<c<15$ )

Boss Level

Boss

a) From $A$ : $h$ ; from $B$ : $h\sqrt3$
b) $h=50\sqrt2\approx70.7$ m
c) $\tan\theta=\frac{50\sqrt2}{50\sqrt3}=\frac{\sqrt6}{3}$ , so $\theta\approx39.2^\circ$

Free cheat sheet

Trigonometric Ratios Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

↓ Build-your-own PDF

Phone card

The essentials on one screen. Save it to your photos.

↓ Phone card (image)

Most testedGiven $\cos\theta=-0.6$ , find all $\theta$ from $0^\circ$ to $360^\circ$ . (card 4)

1Right triangles

$\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\quad\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\quad\tan\theta=\dfrac{\text{opp}}{\text{adj}}$

UseStand at $\theta$ : opposite is the side that doesn't touch it. Missing angle: use $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ .

TipAngles of elevation and depression are measured from the horizontal; bearings, clockwise from north.

Trap $\sin35^\circ=8/x$ gives $x=8/\sin35^\circ\approx13.9$ , not $8\sin35^\circ\approx4.6$ : the hypotenuse is the longest side.

2Special angles

$\theta$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt2}{2}$	$\dfrac{\sqrt3}{2}$	$1$
$\cos$	$1$	$\dfrac{\sqrt3}{2}$	$\dfrac{\sqrt2}{2}$	$\dfrac{1}{2}$	$0$
$\tan$	$0$	$\dfrac{\sqrt3}{3}$	$1$	$\sqrt3$	undef.

TipThe $\sin$ row is $\sqrt0,\sqrt1,\sqrt2,\sqrt3,\sqrt4$ , each over $2$ . The $\cos$ row is the same, backwards.

3Any angle to $360^\circ$

$\sin\theta=\dfrac{y}{r}\quad\cos\theta=\dfrac{x}{r}\quad\tan\theta=\dfrac{y}{x}$

Use $P(x,y)$ on the terminal arm, $r=\sqrt{x^2+y^2}$ . Coterminal angles: $\theta\pm360^\circ$ .

Example $\sin240^\circ$ : QIII, $\beta=60^\circ$ , $\sin$ negative, so $-\sqrt3/2$ .

TrapThe related acute angle (reference angle) is measured to the $x$ -axis, never the $y$ -axis: for $240^\circ$ it is $60^\circ$ , not $30^\circ$ .

4Find $\theta$ from a ratio

Use1. Find $\beta$ from the positive value. 2. CAST: the two quadrants with the right sign. 3. Write $\theta$ in each.

Example $\cos\theta=-0.6$ : $\beta=\cos^{-1}0.6\approx53.1^\circ$ ; $\cos$ is negative in QII and QIII, so $\theta\approx126.9^\circ$ or $233.1^\circ$ .

TrapThe calculator gives one angle, but there are usually two.

5Identities

$\sin^2\theta+\cos^2\theta=1\qquad\tan\theta=\dfrac{\sin\theta}{\cos\theta}$

$\csc\theta=\dfrac{1}{\sin\theta}\;\;\sec\theta=\dfrac{1}{\cos\theta}\;\;\cot\theta=\dfrac{1}{\tan\theta}$

UseTo prove: start from the more complex side, rewrite it in $\sin$ and $\cos$ , and work on that side only.

Example $\sin\theta=-5/13$ in QIV: $\cos^2\theta=144/169$ and $\cos$ is positive in QIV, so $\cos\theta=12/13$ .

Trap $\sec60^\circ=1\div\cos60^\circ=1\div0.5=2$ , not $0.5$ .

6Sine law

$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$

UseAAS, SSA: a side and its opposite angle. ASA: find the third angle first.

TrapEach side pairs with the angle across from it, never an angle next to it.

7The ambiguous case (SSA)

Given $\angle A$ , $a$ , $b$ . Acute $\angle A$ : $h=b\sin A$ .

$a<h$	no triangle
$a=h$	one (right) triangle
$h<a<b$	two triangles
$a\ge b$	one triangle
obtuse $\angle A$ , $a\le b$	no triangle
obtuse $\angle A$ , $a>b$	one triangle

Example $\angle A=30^\circ$ , $a=6$ , $b=10$ : $h=5<6<10$ : $\angle B\approx56.4^\circ$ or $123.6^\circ$ .

Trap $\sin^{-1}$ gives only the acute angle. In SSA, test $180^\circ-\theta$ too.

8Cosine law

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

$\cos C=\dfrac{a^2+b^2-c^2}{2ab}$

UseSAS: the third side. SSS: an angle; find the largest first, so the other two are acute.

ExampleSides $5$ , $7$ , $8$ : $\cos C=(25+49-64)/70=1/7$ , so $C\approx81.8^\circ$ .

Trap $c^2=8^2+10^2-2(8)(10)\cos60^\circ=84$ , not $(164-160)\cos60^\circ$ : multiply first.

93D problems

UseRedraw each triangle flat; a tower meets level ground at $90^\circ$ . Solve the triangle you know most about, then carry the shared side across. Base not right-angled? Sine or cosine law.

TrapA negative $h^2$ means the wrong hypotenuse. Redraw; don't take the absolute value.

Which tool?

There is a right angle SOH CAH TOA

Two sides and the angle between, or three sides Cosine law

A side and its opposite angle Sine law

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Core Skills (K)

K1 Right-Triangle Ratios

K2 Any Angle

K3 Identities

K4 Sine Law

K5 Cosine Law

Real World (A)

A1 Signal on the Highway

A2 Flower Bed

A3 Balloon Spotting

Think It Through (T)

T1 Proof or Not?

T2 The Hidden Angle

Boss Level

Tower Walk

Answer key

Core Skills

Real World

Think It Through

Boss Level

Trigonometric Ratios Cheat Sheet

Unit Summary

Build your own

Phone card

1Right triangles

2Special angles

3Any angle to 360∘360^\circ360∘

4Find θ\thetaθ from a ratio

5Identities

6Sine law

7The ambiguous case (SSA)

8Cosine law

93D problems

Which tool?

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What is VertexMath?

3Any angle to $360^\circ$

4Find $\theta$ from a ratio