Monday
Monday: Tangent From the Special Triangles
Late afternoon, and the old mast throws a long shadow across the dock boards.
Comet walks the tape along it. "Shadow is 4 meters. And Nova's log says the mast is 4 meters tall."
"Height over shadow is 1," Wren says, holding the protractor up to the light. "What do you notice about that?"
"A square corner cut in half," Comet says. "The 45-45-90 triangle. Height equals shadow."
Nova projects the triangle onto the boards. "Would you like a hint? That ratio has its own name."
"Tangent," Wren says. "Opposite over adjacent. Sine over cosine. Tan 45° is 1."
"Then let us measure every special angle," Comet says. "What can we make from two triangles?"
Two triangles that give exact values
Cut a square of side 1 along its diagonal. Each half is a 45-45-90 triangle with legs 1 and 1.
By the Pythagorean theorem the hypotenuse is √(1 + 1) = √2. So sin 45° = 1/√2 = √2/2, cos 45° = √2/2, and tan 45° = 1/1 = 1.
Cut an equilateral triangle of side 2 down the middle. Each half is a 30-60-90 triangle with hypotenuse 2 and short leg 1.
The other leg is √(4 - 1) = √3. So sin 30° = 1/2, cos 30° = √3/2, and tan 30° = 1/√3 = √3/3.
Tangent is sine over cosine
In a right triangle, sine is opposite over hypotenuse and cosine is adjacent over hypotenuse. Divide them and the hypotenuse cancels.
So tangent is opposite over adjacent. For the mast, tangent of the light angle is height over shadow.
For the 60° angle of the 30-60-90 triangle, the opposite leg is √3 and the adjacent leg is 1. So tan 60° = √3.
A solved problem to study
- The crew's readings: mast 4 meters, shadow 4 meters. What is the angle of the light above the dock?
- Tangent of the angle = height ÷ shadow = 4 ÷ 4 = 1.
- Which special angle has tangent 1? In the 45-45-90 triangle, opposite and adjacent are equal, so tan 45° = 1.
- Answer: the light makes a 45° angle with the dock. Check: a 45° triangle with height 4 has a base of 4.
- Later the shadow stretches to 6.9 meters. Then tangent = 4 ÷ 6.9 ≈ 0.58, close to √3/3. The angle is 30°.
Every number here is the crew's own tape reading from Nova's log, not a fact about real shadows or real light.
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | no value |
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| tan 45° = 1 because the two legs of a 45-45-90 triangle are equal. | ? |
| tan 60° = √3 and tan 30° = √3/3. | ? |
| Tangent is adjacent over opposite. | ? |
| The hypotenuse of a 30-60-90 triangle with short leg 1 is √3. | ? |
| π/4 radians is 45°. | ? |
Compare the answer with √2 ≈ 1.41. Then write sin, cos and tan of 45° from your own fold.
Strong start. Tomorrow the angles leave the triangle and walk around the unit circle, with their signs.