Notes
Pythagorean theorem and special triangles
- $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. If a triangle's sides satisfy it, the triangle is right.
- 45-45-90: sides $\displaystyle x, x, x\sqrt{2}$. 30-60-90: sides $\displaystyle x, x\sqrt{3}, 2x$ (shorter leg opposite $30°$).
Trigonometric ratios
- $\displaystyle \sin = \frac{\text{opp}}{\text{hyp}}$, $\displaystyle \cos = \frac{\text{adj}}{\text{hyp}}$, $\displaystyle \tan = \frac{\text{opp}}{\text{adj}}$ (SOH CAH TOA).
- To find a side, pick the ratio that uses the known side and the unknown side. To find an angle, use $\sin^{-1}$, $\cos^{-1}$ or $\tan^{-1}$.
- Use degree mode on your calculator.
Applications
- Angle of elevation and depression are measured from the horizontal. Draw and label a right triangle first.
- Round only at the end of the calculation.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$8^2 + 15^2 = 64 + 225 = 289$ and $\displaystyle \sqrt{289} = 17$.
The hypotenuse is twice the shorter leg, so the shorter leg is $\displaystyle \frac{18}{2} = 9$.
Sine is opposite over hypotenuse: $\displaystyle \frac{5}{13}$.
$\displaystyle \tan 28° = \frac{x}{12}$, so $x = 12 \tan 28° \approx 6.4$.
$\displaystyle \cos\theta = \frac{4}{5} = 0.8$, so $\theta = \cos^{-1}(0.8) \approx 36.9°$.
$\displaystyle \tan 35° = \frac{h}{30}$, so $h = 30 \tan 35° \approx 21$ m.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A kite string is pulled tight and is 80 meters long. It makes a $55°$ angle with the level ground. Assume the string is straight.
(a) How high above the ground is the kite? Round to the nearest tenth.
Show the answer and scoring guide
The height is opposite the $55°$ angle and the string is the hypotenuse: $80 \sin 55° \approx 80(0.8192) \approx 65.5$ m.
Check off the points you earned:
(b) How far along the ground is the kite from the person? Round to the nearest tenth.
Show the answer and scoring guide
The ground distance is adjacent to the angle: $80 \cos 55° \approx 80(0.5736) \approx 45.9$ m.
Check off the points you earned:
(c) Show that your answers are consistent with the Pythagorean theorem.
Show the answer and scoring guide
$65.5^2 + 45.9^2 \approx 4290.3 + 2106.8 = 6397.1$, which is close to $80^2 = 6400$. The small difference comes from rounding.
Check off the points you earned: