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Half Angle: Rule, Formula & Examples

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

If we know at least the cosine of an angle, we can use formulas to find the sine, cosine and tangent of half that angle. These formulas are called half-angle formulas. In this lesson we state half-angle formulas, describe the rules for using them, and present examples.

Half Angles: A Flight School Analogy

Let's say you decide to take up flying. There are two parts to this endeavor: ground school which covers theory and rules, and flight school which uses this information in a simulator and in an actual cockpit. The same might be said for learning math. We learn formulas and rules, then practice by going through examples to show how the theory is applied.

In this lesson on half angles, we will go over the rules, list the formulas and then practice with lots of examples.

Rules for Determining the Correct Sign

The issue with signs arises because the square root has two possible answers. Both the positive and negative of the root are mathematically correct. This doesn't mean that both answers are correct for our half-angle calculations. It means that we have to decide which of the two signs to use.

This task is not that hard if we draw a diagram. There are only four possible cases. We will walk through each of them.

Let's say that the angle in question is between 0o and 90o. This region is called the first quadrant. Draw a triangle in this first quadrant. Label the sides as x, y and r. The hypotenuse r is always positive. Think of number values on the x-axis and the y-axis. In the first quadrant, x > 0 and y > 0. Greater than 0 means positive. Less than 0 means negative.


First Quadrant Sign Rules
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For this representative triangle, sin θ = y/r, cos θ = x/r and tan θ = y/x. In the first quadrant, both x and y are positive. Thus, sin θ > 0, cos θ > 0 and tan θ > 0.

Let's move to the second quadrant.

In the second quadrant, the angles range from 90o to 180o.

Once again, draw a representative triangle in this quadrant. When we label the triangle, we note that x < 0 and y > 0.

Thus, we expect that angles in the second quadrant will have sin θ > 0, cos θ < 0 and tan θ < 0. These conclusions arise by using the definition of the trig functions. If we divide two quantities that have the same sign, the answer will be positive. If the signs are different in our division, the answer will be negative.

The second quadrant sign rules are summarized in the following diagram.


Second Quadrant Sign Rules
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The third quadrant has angles from 180o to 270o. The fourth quadrant has angles from 270o to 360o.

These quadrants and the rules are shown in the following two figures.


Third Quadrant Sign Rules
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Fourth Quadrant Sign Rules
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Listing the Half-Angle Formulas

The half-angle formulas for sine and cosine are as follows.


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For these formulas, we have to be careful in choosing the positive or the negative root.

There are three forms for the half-angle tangent formula.

One of these forms has a square root.


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The other two do not have a square root.


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In the examples that follow, we will use all of these formulas. Angles from each of the four quadrants will be used.

Using the Half-Angle Formulas

Before we start with examples, let's look at a very useful triangle; the 30-60-90 triangle.


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We note that the sine, cosine and tangent are all easily determinable from this 30-60-90 triangle. For example, sin 30o = ½.

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