Trigonometry

Question

Can someone explain Trigonometry to me? Bitesize and Mymaths doesn't help at all!

 

Like you know the SOH CAH TOA thing

and the hypotenuse adjacent and opposite thing

Answer 1

Imagine an overhead projector displaying a 30-60-90 (∠A=30, ∠B=60, ∠C=90), right triangle on a wall.  Now imagine using a tape measure to find the lengths (a, b, and c) of the sides of the triangle on the wall.  Hang with me here...you now start moving the projector farther from the wall, and as you do the triangle gets bigger.  Stop moving the projector, measure sides a, b, and c again.  Pick any 2 measurements from the first measurements you took (while the triangle was small) and divide one by the other.  Do the same for the bigger triangle.

 

a÷b (for the small Δ) = a÷b (for the big Δ)

 

c÷b (for the small Δ) = c÷b (for the big Δ)

 

No matter how far the projector is moved away, these ratios will always be the same.  Trigonometry takes advantage of these relationships to solve problems that would be difficult (or even dangerous) to solve using straightforward methods.  For instance, you could find the height of a flagpole by climbing it with a large tape measure, but that would be very hard to do (as well as dangerous).  Stand a 12-inch ruler up near the flag pole on a sunny day.  When the shadow of the ruler is 6 inches long (half the length of the ruler), measure the length of the shadow that the flag pole casts and the height of the flag pole will be double the length of its shadow, as the height of the ruler is double the length of its shadow.  Much easier, much safer.  I loved teaching Trig because the real-world applications of it were so much more useful to me than the contrived problems for earlier courses (Train A leaves Dallas at time t heading for Houston, while Train B...you remember these).

 

Good luck,

John

Answer 2

Just wanted to add a bit:

 

 

 

As Muhammad says, Trigonometry is all about the relationships between the sides and angles of a triangle.

 

 

 

The very first thing you learn in Trigonometry is often how to find sine, cosine, and tangent given a triangle and all of its side lengths. SOH CAH TOA is a mnemonic device to help you remember how to do that:

 

Sine = Opposite/Hypotenuse (SOH)

Cosine = Adjacent/Hypotenuse (CAH)

Tangent = Opposite/Adjacent (TOA)

 

 

 

Let's say you have a right triangle ABC. Go ahead and draw it, and call the right angle C.

 

The side "across from" vertex C is AB. We call the side across from the right angle the hypotenuse (and we often label it c). It's always the longest side. Say its length is 5.

 

The side "across from" vertex A is the leg BC. We often label it a. It's also adjacent to vertex B. Say its length is 4.

 

The side "adjacent to" vertex A is the leg AC (never the hypotenuse). We often label it b, since it's also across from vertex B. Say its length is 3.

 

Then:

 

sin(A) = opposite/hypotenuse = 4/5

cos(A) = adjacent/hypotenuse = 3/5

tan(A) = opposite/adjacent = 4/3

 

sin(B) = opposite/hypotenuse = 3/5

cos(B) = adjacent/hypotenuse = 4/5

tan(B) = opposite/adjacent = 3/4

 

 

 

You can also use the Pythagorean Theorem (a² + b² = c²) to find a right triangle's third side length if you're given two side lengths. The hypotenuse is always c. So, if one leg is 3 and the hypotenuse is 5, you can find the other leg:

 

3² + b² = 5²

9 + b² = 25

b² = 16

b = 4

 

 

 

These few concepts are the tip of the iceberg when it comes to Trigonometry. It's a whole area of study, so there's no way to explain it all here (just like there's no good way to "explain history" or "explain English" without writing a book). Other basic topics in Trigonometry you can become familiar with include Trigonometric Identities and the Unit Circle.

Answer 3

Simply put, basic trigonometry is the study of the ratios that exist within a right triangle. All right triangles have a right angle, hypotenuse (side opposite the right angle), and two angles that are each less than 90 degrees. These angles all have sides that opposite (far from) them & adjacent (next) to them. The ratio of these sides represent the trig functions sine, cosine, and tangent. The sine of an angle represents the ratio of the opposite side divided by the hypotenuse. The cosine of an angle represents the ratio of the adjacent side divided by the hypotenuse. The tangent of an angle represents the ratio of the opposite side divided by the adjacent side.