Introduction: Trigonometry. This Instructable is originally intended for the ninth students at DIS, but anybody is welcome to learn about Trigonometry. In this Introduction, I will give a general overview of the topic of Trigonometry, tips on how to learn and study well, and then go into more detail. In Mathematics, it is always important to learn how to understand what you are doing, and why you are doing these steps as opposed to just memorizing it. Trigonometry is the study of triangles. In this instructable, I will start basic with naming the sides of the right triangles, the trigonometric functions, and then gradually increase the difficulty so that the reader can eventually see how to tackle these problems, and apply them to real world situations. I will also provide tips on how to study and learn this topic well. This tutorial requires you to already know some basic algebra and geometry.
Step 1: Formulas and Definitions
First Slide: Formulas - Sin = Opp/Hyp, Cos = Adj/Hypotenuse Tan = Opp/Adj
Note: x is the angle we are using to determine the opp, adj, or the hypotenuse. If it were another angle, then the opposite and adjacent would change. Tricks : Soh Cah Toa. You may have seen or heard of this many times. The S in Soh represents the Sine, while the o stands for opposite, and the h stands for hypotenuse.
Definitions: Hypotenuse - The longest side of a right triangle. Opposite to the 90° angle.
Opposite - The side opposite to the angle of reference.
Adjacent - The side that is next to the angle of reference that is not the hypotenuse.
Right Triangle - A triangle with one ninety degree angle.
Step 2: Practice Problems
Second Slide: Steps
a) Please Identify the following sides of the triangles with the appropriate names involving opposite, adjacent, or the hypotenuse.
b) find the sin, cos, tan ratios of the given angle.
c) Solve for side x. (only for the top triangle)
d) use a calculator to find the numerical value of x. (top Triangle)
Tip: Use Pythagoras Theorem To solve for the third unknown side. Opp^2+Adj^2=Hyp^2. Then use algebra to solve for one of these sides.
Answers: Left Triangle- A) Hyp=5m, Adj=4m, Opp=3m B) SinC = ⅗, CosC = ⅘ TanC = ¾
Right Triangle- A) Hyp=x, Adj=unknown side, Opp=2,500.
B) Sin 23 = 2500/x, Cos 23 = unknown side/x, Tan 23 = 2500/unknown side.
C) 1. Sin 23 = 2500/x 2. x Sin 23 = 2500 3. x = 2500/Sin 23. d) Solve with a calculator. Do the same with cos and tan.
Step 3: Finding the Sin and Cos of a Specific Angle
Third Slide: Tricks on how to find the value of the sin, cos, tan of a specific angle.
Exp. Sin 30° = 1 (opposite)/2 (Hypotenuse) so it equals ½ = .5(calculator).
Cos 45° = 1/root 2 = .7071 (Calculator). You can use the pythagorean theorem to check that these are valid right triangles.
There are other examples of finding the ratio defining the trigonometric functions of specific angles. The first step is to find the values of the sides, and then divide them. For most angles, however, you will need a calculator. This step was made to help you understand what the strange numbers and decimals on your calculator mean whenever you find the sin, cos, or tan of an angle.
Step 4: Word Problems
Fourth Slide: These are world problems that are found in real-life situations so that you can put your knowledge into more practical use!
1)You have to first identify the right triangle in this scenario.
2)Then identify the parts ie. adj, hyp, and opp.
3)Find the angle you need to use for your situation. What function will give you the side you need to solve for?
4) Apply the function to that angle, solve for the side, and calculate.
Answer: The angle opposite to the 32° angle is also 32°. Use the tan since the adj is given, and the opposite needs to be found. Tan 32° = ?/325, ? = 325 Tan 32°. The crater is 214.827m deep.
Step 5: Inverse Trigonomic Functions
Fifth Slide: Inverse trigonometric functions.
The goal is to find the measure of an angle given at least two sides. First, you determine the right function to use (tan, sin, and cos) based off of which sides are given (Hyp, Adj, Opp). Then solve for the angle. Exp. Find X. The first step would be to figure out what is given. The opposite (7) and the hypotenuse (25) are known. What trigonometric function involves both the opposite and the hypotenuse? The sine of course! So we create an equation sinx = 7/25. x = arcsin(7/25). Then just type that into your calculator to find the result. The arcsine is just another word for the inverse sin.
Step 6: What We Have Learned
Sixth Slide: Summing it all up. Becoming a better math student.
We have learned what is a right triangle, opp, adj, hyp, sin, cos, tan, how to solve for an unknown side using trigonometry, the pythagorean theorem, values of trigonometric functions for specific angles, applying trigonometry to real world problems, and using the inverse sine to find the value of an angle given the sides. In order to improve, you must practice more math problems. I recommend buying a math book as a source to find a variety of problems, and learn concepts. If you identify your difficulties, be sure to ask for help!