Basic Trigonometry

Basic Trigonometry

When you have a right triangle there are 5 things you can know about it.. • the lengths of the sides (A, B, and C) • the measures of the acute angles (a and b) • (The third angle is always 90 degrees) b C A a B

If you know two of the sides, you can use the Pythagorean theorem to find the other side b C A = 3 a B = 4

And if you know either angle, a or b, you can subtract it from 90 to get the other one: a + b = 90 • This works because there are 180º in a triangle and we are already using up 90º • For example: • if a = 30º • b = 90º – 30º • b = 60º b C A a B

But what if you want to know the angles? • Well, here is the central insight of trigonometry: • If you multiply all the sides of a right triangle by the same number (k), you get a triangle that is a different size, but which has the same angles: k(C) b C b A k(A) a a B k(B)

How does that help us? • Take a triangle where angle b is 60º and angle a is 30º • If side B is 1unit long, then side C must be 2 units long, so that we know that for a triangle of this shape the ratio of side B to C is 1:2 • There are ratios for every shape of triangle! C = 2 60 º A = 1 30º B

But there are three pairs of sides possible! • Yes, so there are three sets of ratios for any triangle • They are mysteriously named: • sin…short for sine • cos…short for cosine • tan…short or tangent • and the ratios are already calculated, you just need to use them

So what are the formulas? B b c a a 900 A b C

Some terminology: • Before we can use the ratios we need to get a few terms straight • The hypotenuse (hyp) is the longest side of the triangle – it never changes • The opposite (opp) is the side directly across from the angle you are considering • The adjacent (adj) is the side right beside the angle you are considering

A picture always helps… • looking at the triangle in terms of angle b b • A is the adjacent (near the angle) C A • B is the opposite (across from the angle) B b Near • C is always the hypotenuse hyp Longest adj opp Across

But if we switch angles… • looking at the triangle in terms of angle a • A is the opposite (across from the angle) C A a • B is the adjacent (near the angle) B Across • C is always the hypotenuse hyp Longest opp a adj Near

Lets try an example • Suppose we want to find angle a • what is side A? • the opposite • what is side B? • the adjacent • with opposite and adjacent we use the… • tan formula b C A = 3 a B = 4

Lets solve it b C A = 3 a B = 4

Where did the numbers for the ratio come from? • Each shape of triangle has three ratios • These ratios are stored your scientific calculator • In the last question, tanθ = 0.75 • On your calculator try 2nd, Tan 0.75 = 36.87 °

Another tangent example… • we want to find angle b • B is the opposite • A is the adjacent • so we use tan b C A = 3 a B = 4

Calculating a side if you know the angle • you know a side (adj) and an angle (25°) • we want to know the opposite side b C A 25° B = 6

Another tangent example • If you know a side and an angle, you can find the other side. b C A = 6 25° B

An application • You look up at an angle of 65° at the top of a tree that is 10m away • the distance to the tree is the adjacent side • the height of the tree is the opposite side 65° 10m

Why do we need the sin & cos? • We use sin and cos when we need to work with the hypotenuse • if you noticed, the tan formula does not have the hypotenuse in it. • so we need different formulas to do this work • sin and cos are the ones! b C = 10 A 25° B

Lets do sin first • we want to find angle a • since we have opp and hyp we use sin b C = 10 A = 5 a B

And one more sin example • find the length of side A • We have the angle and the hyp, and we need the opp b C = 20 A 25° B

And finally cos • We use cos when we need to work with the hyp and adj • so lets find angle b b C = 10 A = 4 a B

Here is an example • Spike wants to ride down a steel beam • The beam is 5m long and is leaning against a tree at an angle of 65° to the ground • His friends want to find out how high up in the air he is when he starts so they can put add it to the doctors report at the hospital • How high up is he?

How do we know which formula to use??? • Well, what are we working with? • We have an angle • We have hyp • We need opp • With these things we will use the sin formula C = 5 B 65°

So lets calculate • so Spike will have fallen 4.53m C = 5 B 65°

One last example… • Lucretia drops her walkman off the Leaning Tower of Pisa when she visits Italy • It falls to the ground 2 meters from the base of the tower • If the tower is at an angle of 88° to the ground, how far did it fall?

First draw a triangle • What parts do we have? • We have an angle • We have the Adjacent • We need the opposite • Since we are working with the adj and opp, we will use the tan formula B 88° 2m

So lets calculate • Lucretia’s walkman fell 57.27m B 88° 2m

What are the steps for doing one of these questions? • Make a diagram if needed • Determine which angle you are working with • Label the sides you are working with • Decide which formula fits the sides • Substitute the values into the formula • Solve the equation for the unknown value • Does the answer make sense?

Two Triangle Problems • Although there are two triangles, you only need to solve one at a time • The big thing is to analyze the system to understand what you are being given • Consider the following problem: • You are standing on the roof of one building looking at another building, and need to find the height of both buildings.

Draw a diagram • You can measure the angle 40° down to the base of other building and up 60° to the top as well. You know the distance between the two buildings is 45m 60° 40° 45m

Break the problem into two triangles. • The first triangle: • The second triangle • note that they share a side 45m long • a and b are heights! a 60° 45m 40° b

The First Triangle • We are dealing with an angle, the opposite and the adjacent • this gives us Tan a 60° 45m

The second triangle • We are dealing with an angle, the opposite and the adjacent • this gives us Tan 45m 40° b

What does it mean? • Look at the diagram now: • the short building is 37.76m tall • the tall building is 77.94m plus 37.76m tall, which equals 115.70m tall 77.94m 60° 40° 37.76m 45m

Basic Trigonometry