Cogito
Geometry · Chapter 5 · Lesson 1
The Three Ratios
Similarity is why they work at all.
10 problems · about 21 minutes · G-SRT.C.6, G-SRT.C.7, G-SRT.C.8
What this lesson teaches
The student defines sine, cosine, and tangent as side ratios and solves right triangles.
- Sine, cosine, and tangent are ratios of sides.
- They depend on the angle alone, because such triangles are all similar.
- Choose the ratio containing the sides you know and want.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — The Side-Splitter Theorem: Third side 16. Midsegment?
Answer 8
Why 8.
Review — Similar Figures and Scale: Sides in ratio 4. Area ratio?
Answer 16
Why 16.
Warm Up
Straightforward practice. Get the method working first.
4 problemsLegs 5 and 12. Hypotenuse?
Answer 13
Why 13.
Why does the sine of an angle not depend on the triangle size?
Answer All such triangles are similar, so the ratios match.
Why Similarity fixes the ratio.
Legs 6 and 8. Hypotenuse?
Answer 10
Why 36 + 64 = 100.
Opposite 3, hypotenuse 5. Sine, as a decimal?
Answer 0.6
Why 3 ÷ 5.
Build It Up
The same ideas with more to keep track of.
2 problemsAdjacent 4, hypotenuse 5. Cosine, as a decimal?
Answer 0.8
Why 4 ÷ 5.
Does scaling a triangle change its sine?
Answer No.
Why Similar triangles share ratios.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Ratio: A right triangle has legs 3 and 4. What is the hypotenuse?
Answer 5
Why 5.
The Tangent: Opposite 6, adjacent 8. What is the tangent, as a decimal?
Answer 0.75
Why 0.75.