Question 69642: I have been studying for an hour and cannot figure out how to do these problems. How do I find the exact value of a trigonometric funtion such as tan135 degrees? Thank you for your help.
Found 2 solutions by stanbon, bucky:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! I have been studying for an hour and cannot figure out how to do these problems. How do I find the exact value of a trigonometric funtion such as tan135 degrees? Thank you for your help.
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Draw a "unit-circle" centered at (0,0) of an xy-coordinate system.
Draw the terminal side of a 135 Degree angle.
From the point where the terminal side meets the circle, draw a perpendicular
to the negative x-axis. That will cut of a segment of the negative x axis.
Do you see the right triangle you have just created?
The terminal side is the hypotenuse; the perpendicular line segment is a side;
the piece of the negative x axis is the third side.
This is a 45-45-90 degree right triangle because you drew a 135 degree angle
as an external angle.
But the hypotentuse is 1 because you drew a "unit-circle".
Using Pythatgoras you can show that the other two sides are each 1/sqrt2.
The coordinates of the point where the terminal side meets the circle
are (-1/sqrt2, 1/sqrt2)
The cosine of 135 degrees is x/r or adj/hyp
Therefore the cos135 = (-1/sqrt2)/1 = -1/sqrt2
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Cheers,
Stan H.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! The easiest way to do it is to it is to punch it in to a scientific calculator. These calculators have sin, cos, and tan keys and cost in the range of $5 to $10. Be sure that the calculator is set for the degrees mode, enter 135 and press the tan key to find that the answer is -1.
Such a calculator is almost a necessity if you are dealing with uncommon angles. By uncommon I mean angles such as 8 degrees, 139.2 degrees, 273 degrees, and so forth. If you don't have a calculator to handle angles such as these, you probably need to resort to a table of trigonometric functions such as the "old-time" mathematicians used when they were struggling along.
Now, as to the particular problem you presented. As you read the following, it probably will be more understandable if you sketch the following thoughts onto a Cartesian coordinate system so you have a better picture of what is going on.
135 degrees is a fairly common angle. If you draw it onto a Cartesian coordinate system, it is an angle that is located in the second quadrant -- the upper left hand quarter of the graph. This angle is represented by a line that originates at the origin and slopes up and to the left such that the angle between this line and the negative x-axis is 45 degrees.
Notice that in addition to being at 45 degrees with the negative x-axis, the angle between this line and the positive y-axis if 45 degrees. If you start an angle measurement from the positive x-axis to the sloped line, measuring in the counter-clockwise direction, you will see that the angle is 135 degrees (90 degrees to the +y-axis plus 45 degrees from the +y-axis to the sloped line).
Recall from geometry that a right triangle (the kind most commonly used in trigonometry) that contains a 45 degree angle has the other acute angle of 45 degrees. (45 + 45 + 90 = 180 degrees for the triangle.) Also recall from geometry that the sides of such a 45 degree triangle are in the ratio of 1 to 1 to sqrt(2). These dimensions come into play during the next couple of steps.
Let's examine the characteristics of the line you drew from the origin and sloped it up and to the left at an angle of 45 degrees with the negative x-axis can be described this way: go to the left on the negative x-axis 1 unit from the origin. From that point (-1, 0) go vertically up one unit. The point that you are at now is (-1, +1) and it lies on the sloped line from the origin. [The distance from the origin to the point (-1, +1) is sqrt(2).]
[Note you could also multiply everything by 10. Go to the left on the negative x-axis 10 units from the origin to the point (-10, 0). From that point go vertically upward ten units to the point (-10, + 10). That point is on the line from the origin and the distance from the origin to the point (-10, +10) is equal to 10 times the square root of 2. This whole exercise just makes the sketch a little bigger.]
Looking on your sketch you should see that the three points (0,0), (-1,0) and (-1, +1) form a right triangle. [The same can be said for the scaled up points of (0,0), (-10,0) and (-10, +10)]
Now recall the definition of the tangent. It is the length of the side opposite to the angle divided by the length of the side adjacent to the angle. The angle is always measured between the sloped line and the x-axis. So in this case the side opposite the 45 degree angle is +1, the distance upward in the y direction, and the length of the side adjacent is -1, the distance from the origin to the left. Therefore:
Just for grins, how about finding sin(135). From the definition of sine you know that it equals the side opposite divided by the hypotenuse. Just as before, the side opposite is the length from the point (-1,0) to the point (-1, +1), a distance of +1 unit. The hypotenuse is the length from the origin to the point (-1, +1) which is the square root of 2. The hypotenuse is defined as always being a positive number. So sin(135):
which can be found by approximating the square root of 2 as 1.4142136 to get:
and this divides out to be Sin(135) = +0.707107
This has probably been somewhat confusing, but hang with it and it will eventually get clear to you. If you have further questions about it, post a follow up message.
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