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Sin/Cos/Tan

Introduction

Sin/Cos/Tan is a very basic form of trigonometry that allows you to find the lengths and angles of right-angled triangles. A very easy way to remember the three rules is to  to use the abbreviation SOH CAH TOA. It is very important that you know how to apply this rule.

Using Sin/Cos/Tan to find Lengths of Right-Angled Triangles

Before you start finding the length of the unknown side, you need to know two things: 1 angle and 1 other length. Then you should annotate the triangle with Opposite (the side opposite to the known angle), Hypotenuse (side opposite the right angle) and Adjacent (the remaining side). Then you can start solving the problem. First you need to see which formula you have to use. To do this see which side you know and which one you need to find. In this case, we need to find the opposite and know the adjacent and so we have to use the Tan formula. Substitute the values into the formula as shown on the right. Then solve the formula by multiplying both sides by 8 and then finding 8 times tan(43). This gives us the solution. The same method is also used for the Cos and Sin formulas.

Exception to this Method:

There is an exception to this method which is when the unknown side is at the bottom of the fraction. Let’s imagine that in the previous example, the unknown side is the adjacent and the opposite is 8 cm long. On the right you can see how the method would change. Since we cannot solve the equation by multiplying by the denominator of the fraction, we have to swap the denominator (?) with the other side of the equal side (tan43). Using this method we are able to solve the equation. Have a look at the examples underneath using both of these methods.
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Example 1 In this example, we have used the Sin formula to find the missing length. We start of be substituting the values into the formula. Since the denominator of the fraction is the unknown length, we have to switch it with the other side of the equal sign in order to solve the equation. The final answer is 5.82 (3sf).
Example 2 In this example, we had to use the Cos formula to find the missing length. We simply substituted the values into the formula and then multiplied both sides by the denominator of the fraction to solve for ?. The final answer is 3.52cm (3sf).

Using Sin/Cos/Tan to find Angles of Right-Angled Triangles

We can also use Sin/Cos/Tan to find missing angles in right-angled triangles. To do this, we have to use Sin/Cos/Tan to the power of -1. Start of by substituting the values into the formula as on the right. To find the angle, you then have to use Sin/Cos/Tan to the power of -1. The easiest way is to do this as done in the example is to write the formula as ?=sin^-1(4/7). Then use a calculator to solve sin^-1(4/7) and to find the actual size of the angle which is 34.8°. Sin/Cos/Tan to the power of -1 is also known as inverse Sin/Cos/Tan. Here are some examples that will help you to understand this important concept.
Example 1 In this example, we have used inverse Sin to find the missing angle. We have simply substituted the values into the Sin formula and then used the inverse Sin function on the calculator to find the value of ?. The unknown angle is 26.6°.
Example 2 To solve this problem we have to use inverse cos function of the calculator. We started off by substituting the known values into the Cos formula before using the inverse Cos function to find the size of the angle. The unknown angle is 63.6°.

More Advanced Trigonometry

Now that you know how to work with the basic Sin/Cos/Tan formulas, you should learn how to use sine and cosine rule as they will allow you to find missing lengths and angles in non right-angled triangles. Please share this page if you like it or found it helpful!
Sine Rule 31.0 TRIGONOMETRY 31.2 SINE RULE 31.3 COSINE RULE
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Chapter 31.1:  Learning Outcomes Students will learn how to use sin, cos and tan in order to find angles and sides of right-angled triangles!

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