The Tangent Ratio

Worked example

1. 1

   $\tan x=\frac{4.5}{8}$ — The rise 4.5 is opposite the angle and the 8 across is adjacent to it. Tan is opposite over adjacent, so 4.5 goes on top.

2. 2

   $\tan x=0.5625$ — Divide 4.5 by 8 on the calculator to get the ratio.

3. 3

   $x={\tan }^{-1}(0.5625)$ — Inverse tan undoes tan, so it turns the ratio back into an angle.

4. 4

   $x=29.358{\dots }^{\circ }$ — Type it in with the calculator in degrees mode and keep all the digits.

5. 5

   $x=29.4^{\circ }$ — Round to 1 d.p. The digit after the first decimal place is 5, so round up.

Worked example

1. 1

   $\tan 38^{\circ }=\frac{x}{7}$ — The 7 cm side touches the angle, so it is the adjacent. $x$ is across from the angle, so it is the opposite and goes on top.

2. 2

   $x=7\times \tan 38^{\circ }$ — $x$ is divided by 7, so multiply both sides by 7 to get $x$ alone.

3. 3

   $x=5.469\dots$ — Type 7 × tan 38 into the calculator in degrees mode.

4. 4

   $x=5.47\text{ cm}$ — Round to 3 s.f. The next digit is 9, so round the 6 up to 7.

Worked example

1. 1

   $\tan 52^{\circ }=\frac{9}{x}$ — The 9 cm side is opposite the angle. $x$ touches the angle, so it is the adjacent and goes on the bottom.

2. 2

   $x\times \tan 52^{\circ }=9$ — Multiply both sides by $x$ to bring it off the bottom.

3. 3

   $x=9÷\tan 52^{\circ }$ — $x$ is multiplied by $\tan 52^{\circ }$, so divide both sides by $\tan 52^{\circ }$.

4. 4

   $x=7.031\dots$ — Work out 9 ÷ tan 52 on the calculator in degrees mode.

5. 5

   $x=7.03\text{ cm}$ — Round to 3 s.f. The next digit is 1, so the 3 stays.