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1. Locate the point on the coordinate plane and connect it to the origin, using a straight line.
2. Draw a perpendicular line connecting the given point to the x-axis, creating a right triangle.
3. Find the length of the hypotenuse r by using the distance formula or the Pythagorean theorem.
4. Evaluate the trig function values, using their alternate definitions.
All you need to do is plug it in.
1. Say, for example, that you're asked to evaluate all six trig functions of the angle between the positive x-axis and the line segment joining the origin to the point in the plane (–4, –6). The line segment moving from this point to the origin is your hypotenuse and is now called the radius r (as you see in the figure).
2. The legs of the right triangle are –4 and –6. Don't let the negative signs scare you; the lengths of the sides are still 4 and 6. The negative signs just reveal the location of that point on the coordinate plane.
3. The distance you want to find is the length of r from Step 1. Using the distance formula between (x, y) and the origin (0, 0), you get
Remember that this equation implies the principal or positive root only, so the hypotenuse for these point-in-the-plane triangles is always positive.
For this example, you get
which simplifies to
4. With the labels from the figure, you get the following formulas:
4. With the labels from the figure, you get the following formulas:
Substitute the numbers from the example in the figure to pinpoint the trig values:
• Simplify first:
• Then rationalize the denominator:
• Simplify first:
• Then rationalize:
• This answer simplifies to 3/2.
Notice that the rules of trig functions and their reciprocals still apply. For example, if you know
you automatically know
because they're reciprocals.