Trigonometric Ratios on the Unit Circle
Recall special right triangles from Geometry. In a triangle, the sides are in the ratio .
In an isosceles triangle , the congruent sides and the hypotenuse are in the ratio .
In a triangle, the sides are in the ratio .
Now let’s make the hypotenuse equal to 1 in each of the triangles so we’ll be able to put them inside the unit circle. Using the appropriate ratios, the new side lengths are:
Using these triangles, we can evaluate sine, cosine and tangent for each of the angle measures.
These triangles can now fit inside the unit circle.
Putting together the trigonometric ratios and the coordinates of the points on the circle, which represent the lengths of the legs of the triangles, , we can see that each point is actually , where is the reference angle. For example, is the – coordinate of the point on the unit circle in the triangle with reference angle . By reflecting these triangles across the axes and finding the points on the axes, we can find the trigonometric ratios of all multiples of and (or radians).
Let's solve the following problems using the unit circle.
- Find .
Find on the unit circle and the corresponding point is . Since each point on the unit circle is .
- Find .
This time we need to look at the ratio . We can use the unit circle to find and . Now, .
Quadrants in a Unit Circle
Another way to approach these exact value problems is to use the reference angles and the special right triangles. The benefit of this method is that there is no need to memorize the entire unit circle. If you memorize the special right triangles, can determine reference angles and know where the ratios are positive and negative you can put the pieces together to get the ratios. Looking at the unit circle above, we see that all of the ratios are positive in Quadrant I, sine is the only positive ratio in Quadrant II, tangent is the only positive ratio in Quadrant III and cosine is the only positive ratio in Quadrant IV.
Keeping this diagram in mind will help you remember where cosine, sine and tangent are positive and negative. You can also use the pneumonic device - All Students Take Calculus, or ASTC, to recall which is positive (all the others would be negative) in which quadrant.
The coordinates on the vertices will help you determine the ratios for the multiples of or .
Now, let's find the exact values for the following trigonometric functions using the alternative method.
First, we need to determine in which quadrant the angles lies. Since is between and it will lie in Quadrant II. Next, find the reference angle. Since we are in QII, we will subtract from to get . We can use the reference angle to find the ratio, . Since we are in QII where only sine is positive, .
This time we will need to work in terms of radians but the process is the same. The angle lies in QIV and the reference angle is . This means that our ratio will be negative. Since .
The angle represents more than one entire revolution and it is equivalent to . Since our angle is a multiple of we are looking at an angle on an axis. In this case, the point is . Because , which is undefined. Thus, is undefined.
Earlier, you were asked to find the exact values of the following trigonometric ratios.
First, we need to determine in which quadrant the angle lies. Since is between and it will lie in Quadrant II. Next, find the reference angle. Since we are in QII, we will subtract from to get . We can use the reference angle to find the ratio, . Since we are in QII where only sine is positive, .
In problem #2 above we established that the angle lies in QIV and the reference angle is . This means that the tangent ratio will be negative. Since .
Find the exact trigonometric ratios. You may use either method.
has a reference angle of in QI. and since cosine is positive in QI, .
is coterminal to which has coordinates (0, 1). So which is undefined.
has a reference angle of in QI. and since sine is positive in QI, .
is coterminal to in QIV. and since tangent is negative in QIV, .
is coterminal to in QII. and since cosine is negative in QII, .
Find the exact values for the following trigonometric functions.
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 13.7.