Three ratios that measure an angle
Tonight's idea
Draw a right triangle with a 35° angle. Draw another, any size. Both have angles 35°, 90° and 55°, so they are the same shape: one is a scaled copy of the other. Scaling multiplies every side by the same factor, so the ratio of any two sides stays the same. (The same argument made slope constant in grade 8.)
Name the sides as the angle sees them. The hypotenuse faces the right angle. The opposite side is across from the angle. The adjacent side is next to it.
That gives three ratios which depend only on the angle \(\theta\) (theta, a Greek letter often used for angles):
\(\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}\)
The names are short for sine, cosine and tangent. A calculator in degree mode knows them: \(\sin 35^\circ \approx 0.574\), \(\cos 35^\circ \approx 0.819\), \(\tan 35^\circ \approx 0.700\). So whenever a right triangle has a 35° angle, the opposite side is 0.574 of the hypotenuse. Know the angle and one side, and you can find the rest.
Watch one: the height of a tree
You stand 20 m from a tree on level ground. The line from your feet to its top makes 35° with the ground. How tall is the tree?
The ground is adjacent to the 35° angle and the tree is opposite. The ratio linking those two is tan: \(\tan 35^\circ = \dfrac{h}{20}\), so \(h = 20 \times \tan 35^\circ \approx 20 \times 0.700 = 14.0\) m.
How long is the line to the top? It is the hypotenuse: \(\cos 35^\circ = \dfrac{20}{d}\), so \(d = \dfrac{20}{\cos 35^\circ} \approx 24.4\) m.
Drag the angle and resize the triangle; the ratios only notice the angle: mathdetours.com/demos/trigtriangle.html. Next year the angle is allowed past 90°: SOHCAHTOA grows up.
Your turn
- A right triangle has sides 3, 4 and 5. The angle \(\theta\) is opposite the side of length 3. Write \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\) as fractions.
- Find the side, to two decimal places. (a) Hypotenuse 12, angle 35°: the opposite side. (b) Adjacent 8, angle 50°: the opposite side. (c) Hypotenuse 20, angle 70°: the adjacent side.
- No calculator. Use the two triangles from the last sheet (sides \(1, 1, \sqrt2\) and \(1, \sqrt3, 2\)) to write exact values of \(\sin 30^\circ\), \(\cos 30^\circ\), \(\tan 45^\circ\) and \(\tan 60^\circ\).
- (a) A playground slide makes 35° with the ground and its top is 3 m high. How long is the slide? (b) Why can \(\sin\theta\) and \(\cos\theta\) never be bigger than 1 in a right triangle, while \(\tan\theta\) can? (c) For which angle is \(\tan\theta\) exactly 1?
Stretch: two facts that hold for every angle
Call the sides \(o\), \(a\) and \(h\). (a) Show that \(\sin\theta \div \cos\theta = \tan\theta\). (b) Use Pythagoras to show that \((\sin\theta)^2 + (\cos\theta)^2 = 1\). Check both on a calculator with \(\theta = 35^\circ\).
Answers
- Opposite 3, adjacent 4, hypotenuse 5: \(\sin\theta = \tfrac35\), \(\cos\theta = \tfrac45\), \(\tan\theta = \tfrac34\).
- (a) \(12 \times \sin 35^\circ \approx 6.88\). (b) \(8 \times \tan 50^\circ \approx 9.53\). (c) \(20 \times \cos 70^\circ \approx 6.84\).
- In the \(1, \sqrt3, 2\) triangle the side opposite 30° is 1 and the side opposite 60° is \(\sqrt3\). So \(\sin 30^\circ = \tfrac12\), \(\cos 30^\circ = \dfrac{\sqrt3}{2}\), \(\tan 60^\circ = \sqrt3\). In the \(1, 1, \sqrt2\) triangle, \(\tan 45^\circ = \tfrac11 = 1\).
- (a) The height is opposite and the slide is the hypotenuse: \(\sin 35^\circ = \dfrac{3}{L}\), so \(L = \dfrac{3}{\sin 35^\circ} \approx 5.23\) m. (b) The hypotenuse is the longest side, so a leg divided by it is less than 1. For tan, the opposite leg can be longer than the adjacent one. (c) 45°, where the two legs are equal.
Stretch. (a) \(\dfrac{o}{h} \div \dfrac{a}{h} = \dfrac{o}{h} \times \dfrac{h}{a} = \dfrac{o}{a} = \tan\theta\). (b) \(\dfrac{o^2}{h^2} + \dfrac{a^2}{h^2} = \dfrac{o^2 + a^2}{h^2} = \dfrac{h^2}{h^2} = 1\), since \(o^2 + a^2 = h^2\). Check: \(0.574 \div 0.819 \approx 0.70\), and \(0.574^2 + 0.819^2 \approx 0.329 + 0.671 = 1.000\).