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We can use trig to slam-dunk the opposing team in basketball or do real damage to a tennis challenger in a singles match. We can figure out the ideal distance between the basket and the foul line by doing a trig calculation. Hint: the distance is one side of a triangle, and if we do the Pythagorean thing—more on that later—we can figure it all out.
Behold your mathematical journey:
Right angles → Trigonometry → Trigonometric Functions → Calculus → Physics → The Meaning of Life
Okay, understanding right angles might not help us understand the meaning of life. Knowing about right triangles does help us understand the physical world, though.
Trigonometric functions (also called trig functions by the all cool kids) are all about the angles. What did you expect—Benjamins? With trig, we can study the relationship between these right triangles and the sides that sit opposite to them. (Warning: it's hot and heavy).
Let's get this trigonometry thing started.
Take a peek at ∆ABC:
Look! It's a triangle with legs! Okay, maybe it's not wearing jeans. But it still has legs. Angle C is the right angle, and side c is the hypotenuse.
Sides a and b are the legs.
Remember from geometry that the side opposite an angle gets the same name, just a lower case letter.
This means side a is opposite angle A and side b is opposite angle B. The hypotenuse, usually designated by letter c, is always opposite the right angle C.
A long time ago, when philosophy ruled and Socrates drank hemlock, a brainiac named Pythagoras proved that for right triangles:
a2 + b2 = c2
Subsequent admirers and right-angle folks know this equation as the Pythagorean Theorem. Sides a and b are the legs, and side c is the hypotenuse (the one opposite the right angle).
This means that if we know the length of two sides of a right triangle, we have the third side in the bag. Just remember how to use your squares and square roots and you're good to go.
In the following triangle, how long is side c?
First, jot down the Pythagorean Theorem.
a2 + b2 = c2
We're looking for the hypotenuse, which means we know a and b. Next, plug in 4 for a and 3 for b.
42 + 32 = c2
16 + 9 = c2
c2 = 25
Take the square root of both sides. (We're rooting for you.)
c = 5
There we go. Missing side = found.
What's the length of side a in this triangle?
Rearrange the Pythagorean Theorem to solve for a2.
a2 + b2 = c2
a2 = c2 – b2
Plug in b = 6 and c = 10.
a2 = 102 – 62
a2 = 100 – 36
a2 = 64
Take the square root of both sides for the finishing touch.
a = 8
How long is side b in this triangle?
Arrange the Pythagorean Theorem in terms of b2.
a2 + b2 = c2
b2 = c2 – a2
Plug 'em in, then solve for b with some clever square-rooting.
b2 = 132 – 122
b2 = 169 – 144
b2 = 25
b = 5
We've had our fun with triangles and legs, Pythagoras and his invention, and exponents and square roots. But seriously, that stuff isn't at the core of trig.
Real trig rocks to the tune of sine, cosine, and tangent.
The three basic trigonometric ratios are sine, cosine, and tangent.
Stay with us. We aren't speaking in a differently language (unless you consider Latin to be a different language).
Their abbreviations/nicknames are sin, cos, and tan. A good way to remember their definitions is this weird acronym:
Let's see how this guy works with a right triangle:
SOH stands for Sine equals the Opposite side over the Hypotenuse:
CAH stands for Cosine equals the Adjacent side over the Hypotenuse:
TOH stands for Tangent equals the Opposite side over the Adjacent side:
Here's an important little factoid: these ratios only work with right triangles. If there's not a right angle in your triangle, all bets are off.
If a = 8 and b = 15 in the following right triangle, find the sine, cosine, and tangent of angle A.
First, use a2 + b2 = c2 to find c, the hypotenuse.
c2 = 82 + 152
c2 = 64 + 225
c2 = 289
c = 17
Now let's plug c = 17 into our relationships (oh no, not those again) for sin, cos, and tan.
In the following triangle, find the sine, cosine, and tangent of angle B.
If the B is throwing you for a loop, don't worry. You still apply the ratios the same way. Sine is still opposite side over hypotenuse, and so on.
But first, we need that missing leg. We know the other leg and the hypotenuse, so we can use Pythagoras to find a.
a2 = c2 – b2
a2 = 292 – 202
a2 = 841 – 400
a2 = 441
a = 21
In relation to angle B, 20 is the opposite side, 21 is the adjacent side, and 29 is the hypotenuse. Throw those into the trig ratios and we're done.
What are the sine, cosine, and tangent of angle A?
First find c.
c2 = 12 + 12
c2 = 1 + 1
c2 = 2
Now just take the square root of both sides.
Finishing move: apply the trig ratios.
Hint: Don't forget to rationalize your denominators. In other words, make sure there aren't any radical signs on the bottoms of your fractions. For a refresher on this little calculation, go here.