Choosing the Right Trigonometric Ratio

Steps

1. 1

   1. Label the sides hyp, opp and adj from the marked angle.

2. 2

   2. Tick the side you know and the side you want.

3. 3

   3. Pick the ratio that uses those two sides.

Worked example

1. 1

   $\text{know hyp},\text{want adj}$ — 18.6 cm is the hypotenuse. The side we want touches the angle, so it is the adjacent. Tick those two.

2. 2

   $\cos 52^{\circ }=\frac{x}{18.6}$ — Adjacent and hypotenuse are the CAH pair, so use cosine.

3. 3

   $x=18.6\times \cos 52^{\circ }$ — $x$ is divided by 18.6, so multiply both sides by 18.6.

4. 4

   $x=11.451\dots$ — Type 18.6 × cos 52 into the calculator in degrees mode.

5. 5

   $x=11.5\text{ cm}$ — Round to 3 s.f. The next digit is 5, so round up.

Worked example

1. 1

   $\text{know adj},\text{want hyp}$ — 7 cm touches the angle and is not the hypotenuse, so it is the adjacent. The side we want is the hypotenuse. Tick those two.

2. 2

   $\cos 40^{\circ }=\frac{7}{x}$ — Adjacent and hypotenuse are the CAH pair, so use cosine.

3. 3

   $x\times \cos 40^{\circ }=7$ — Multiply both sides by $x$ to bring it off the bottom.

4. 4

   $x=7÷\cos 40^{\circ }$ — $x$ is multiplied by $\cos 40^{\circ }$, so divide both sides by $\cos 40^{\circ }$.

5. 5

   $x=9.138\dots =9.14\text{ cm}$ — Round to 3 s.f. The hypotenuse is the longest side, and 9.14 is longer than 7, so it makes sense.

Worked example

1. 1

   $\text{know opp and adj}$ — 4.1 cm is across from the angle and 6.3 cm touches it. Neither is the hypotenuse, so tick opposite and adjacent.

2. 2

   $\tan x=\frac{4.1}{6.3}$ — Opposite and adjacent are the TOA pair, so use tangent.

3. 3

   $\tan x=0.6507\dots$ — Divide 4.1 by 6.3 on the calculator.

4. 4

   $x={\tan }^{-1}(0.6507\dots )$ — Inverse tan turns the ratio back into an angle.

5. 5

   $x=33.05{\dots }^{\circ }$ — Type it in with the calculator in degrees mode.

6. 6

   $x=33.1^{\circ }$ — Round to 1 d.p. The next digit is 5, so round up.

Worked example

1. 1

   $\tan 25^{\circ }=\frac{CD}{10}$ — Triangle $ACD$ has a right angle at $D$. $AD$ touches the 25° angle and $CD$ is across from it, so use tangent.

2. 2

   $CD=10\times \tan 25^{\circ }=4.663\dots$ — Multiply both sides by 10. Keep every digit on the calculator.

3. 3

   $BD=4.663\dots +6=10.663\dots$ — $C$ is on $BD$, so $BD$ is $CD$ plus $BC$.

4. 4

   $\tan BAD=\frac{10.663\dots }{10}$ — Now use triangle $ABD$: $BD$ is opposite angle $BAD$ and $AD$ is adjacent to it.

5. 5

   $BAD=46.84{\dots }^{\circ }$ — Inverse tan turns the ratio back into an angle.

6. 6

   $BAC=46.84-25=21.8^{\circ }$ — Angle $CAD$ sits inside angle $BAD$, so take it off to leave angle $BAC$.

Worked example

1. 1

   $\text{draw the line from the top to the base}$ — The line from the top corner to the middle of the base meets it at 90°. That makes a right-angled triangle.

2. 2

   $\text{half the base}=10÷2=5$ — The line cuts the base in half, so the side touching $x$ is 5 cm.

3. 3

   $\text{adj}=5,\text{hyp}=8$ — The 5 cm side touches $x$, so it is the adjacent. The 8 cm side faces the right angle, so it is the hypotenuse.

4. 4

   $\cos x=\frac{5}{8}$ — Adjacent and hypotenuse are the CAH pair, so use cosine.

5. 5

   $x={\cos }^{-1}(0.625)$ — Inverse cos turns the ratio back into an angle.

6. 6

   $x=51.31\dots =51.3^{\circ }$ — Round to 1 d.p. The next digit is 1, so the 3 stays.