The Cosine Function
Look again at Fig. 91. Imagine, once again, a ray OP from the origin outward through point P on the circle, pointing right along the x axis, and then rotating in a counterclockwise direction.
What happens to the value of x _{0} during one complete revolution of the ray? It starts out at x _{0} = 1, then decreases until it reaches x _{0} = 0 when θ = 90° = π/2 rad. After that, x _{0} continues to decrease, getting down to x _{0} = −1 when θ = 180° = π rad. As P continues counterclockwise around the circle, x _{0} begins to increase again; at θ = 270° = 3 π /2 rad, the value gets back up to x _{0} = 0. After that, x _{0} increases further until, when P has gone completely around the circle, it returns to x _{0} = 1 for θ = 360° = 2π rad.
The value of x _{0} is defined as the cosine of the angle θ . The cosine function is abbreviated cos. So we can write this:
cos θ = x _{0}
Fig. 91 . The unit circle is the basis for the trigonometric functions. some point P = ( x _{0} , y _{0} ). Then we can define three mathematical functions, called circular functions , of the angle θ in a simple way.
The Tangent Function
Once again, refer to Fig. 91. The tangent (abbreviated tan) of an angle θ is defined using the same ray OP and the same point P = ( x _{0} , y _{0} ) as is done with the sine and cosine functions. The definition is:
tan θ = y _{0} / x _{0}
Because we already know that sin θ = y _{0} and cos θ = x _{0} , we can express the tangent function in terms of the sine and the cosine:
tan θ = sin θ /cos θ
This function is interesting because, unlike the sine and cosine functions, it “blows up” at certain values of θ . Whenever x _{0} = 0, the denominator of either quotient above becomes zero. Division by zero is not defined, and that means the tangent function is not defined for any angle θ such that cos θ = 0. Such angles are all the odd multiples of 90° ( π /2 rad).
Fig. 91 . The unit circle is the basis for the trigonometric functions. some point P = ( x _{0} , y _{0} ). Then we can define three mathematical functions, called circular functions , of the angle θ in a simple way.
PROBLEM 91
What is tan 45°? Do not perform any calculations. You should be able to infer this without having to write down a single numeral, and without using a calculator.
SOLUTION 91
Draw a diagram of a unit circle, such as the one in Fig. 91, and place ray OP such that it subtends an angle of 45° with respect to the x axis. That angle is the angle of which we want to find the tangent. Note that the ray OP also subtends an angle of 45° with respect to the y axis, because the x and y axes are perpendicular (they are oriented at 90° with respect to each other), and 45° is exactly half of 90°. Every point on the ray OP is equally distant from the x and y axes; this includes the point ( x _{0} , y _{0} ). It follows that x _{0} and y _{0} must be the same, and neither of them is zero. From this, we can conclude that y _{0} / x _{0} = 1. According to the definition of the tangent function, therefore, tan 45° = 1.
Practice problems for these concepts can be found at: Vectors And Cartesian ThreeSpace Practice Test.
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