1. Trigonometry
Review

2. Instant Trig Trigonometry is math, so many people find it scary
However, 95% of all the “trig” you’ll ever need to know can be covered in 15 minutes

3. Angles add to 180° The angles of a triangle always add up to 180° 20°
120°
30°
44°
68°
20°
30°
180°
+ 130°
44°
68°
+ 68°
180°

4. Right triangles We only care about right triangles
A right triangle is one in which one of the angles is 90°
Here’s a right triangle:
Here’s the angle
we are looking at
Here’s the
right angle
hypotenuse
opposite
adjacent

5. Right triangles Here’s the angle we are looking at Here’s the
right angle
hypotenuse
opposite
adjacent
We call the longest side the hypotenuse
We pick (or told to use) one of the other angles--not the right angle
We name the other two sides relative to that angle

6. The Pythagorean Theorem
If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse
adj2 + opp2 = hyp2
= 52, or = 25

7. hyp
adj
opp
There are few triangles with integer sides that satisfy the Pythagorean formula
3-4-5 and its multiples (6-8-10, etc.) are the best known
and its multiples form another set
= 169

8. Ratios Each of these six ratios has a name (and an abbreviation)
Three ratios are most used:
sine = sin = opp / hyp
cosine = cos = adj / hyp
tangent = tan = opp / adj
hypotenuse
adjacent
opposite
The ratios depend on the shape of the triangle (the angles) but not on the size
hypotenuse
adjacent
opposite

9. Using the ratios
With these functions, if you know an angle (in addition to the right angle) and the length of a side, you can compute all other angles and lengths of sides
hypotenuse
adjacent
opposite
To calculate a side ratio take the tan/sin/cos of the angle
Tan θ = opp/adj
To calculate an angle take the tan-1/sin-1/cos-1 of the side ratio
θ = tan-1 (opp/adj)

10. Using the ratios hypotenuse opposite adjacent
If you know the angle marked in red and you know the length of the adjacent side, then
tan A = opp / adj, so length of opposite side is given by opp = adj * tan A
cos A = adj / hyp, so length of hypotenuse is given by hyp = adj / cos A

11. The Hard Part Here’s the part I’ve always found the hardest:
Which ratio do you use and when?
It depends on what you are given!
If given opp + adj use tan
If given opp + hyp use sin
If given adj + hyp use cos
sin A = opp / hyp
cos A = adj / hyp
tan A = opp / adj
hypotenuse
adjacent
opposite

12. Mnemonics from wikiquote
The formulas for right-triangle trigonometric functions are:
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Mnemonic for those formulas are:
Some Old Horse Caught Another Horse Taking Oats Away
Saints On High Can Always Have Tea Or Alcohol
SOH CAH TOA

13. Solve for angle A and C

14. Solve for A and B

15. Find x in mm’s

16. Find x in m’s

17. Solve each Triangle

18. Solve each Triangle