1.4: Trigonometric Functions of Any Angle
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To define the trigonometric functions of any angle - including angles less than 0∘ or greater than 360∘ - we need a more general definition of an angle. We say that an angle is formed by rotating a ray →OA about the endpoint O (called the vertex), so that the ray is in a new position, denoted by the ray →OB. The ray →OA is called the initial side of the angle, and →OB is the terminal side of the angle (see Figure 1.4.1(a)).

We denote the angle formed by this rotation as ∠AOB, or simply ∠O, or even just O. If the rotation is counter-clockwise then we say that the angle is positive, and the angle is negative if the rotation is clockwise (see Figure 1.4.1(b)).
One full counter-clockwise rotation of →OA back onto itself (called a revolution), so that the terminal side coincides with the initial side, is an angle of 360∘; in the clockwise direction this would be −360∘. Not rotating →OA constitutes an angle of 0∘. More than one full rotation creates an angle greater than 360∘. For example, notice that 30∘ and 390∘ have the same terminal side in Figure 1.4.2, since 30+360=390.

We can now define the trigonometric functions of any angle in terms of Cartesian coordinates. Recall that the xy-coordinate plane consists of points denoted by pairs (x,y) of real numbers. The first number, x, is the point's x coordinate, and the second number, y, is its y coordinate. The x and y coordinates are measured by their positions along the x-axis and y-axis, respectively, which determine the point's position in the plane. This divides the xy-coordinate plane into four quadrants (denoted by QI, QII, QIII, QIV), based on the signs of x and y (see Figure 1.4.3(a)-(b)).

Now let θ be any angle. We say that θ is in standard position if its initial side is the positive x-axis and its vertex is the origin (0,0). Pick any point (x,y) on the terminal side of θ a distance r>0 from the origin (see Figure 1.4.3(c)). (Note that r=√x2+y2. Why?) We then define the trigonometric functions of θ as follows:
sinθ = yrcosθ = xrtanθ = yx
cscθ = rysecθ = rxcotθ = xy
As in the acute case, by the use of similar triangles these definitions are well-defined (i.e. they do not depend on which point (x,y) we choose on the terminal side of θ). Also, notice that |sinθ|≤1 and |cosθ|≤1, since |y|≤r and |x|≤r in the above definitions.
Notice that in the case of an acute angle these definitions are equivalent to our earlier definitions in terms of right triangles: draw a right triangle with angle θ such that x=adjacent side, y=opposite side, and r=hypotenuse. For example, this would give us sinθ=yr=oppositehypotenuse and cosθ=xr=adjacenthypotenuse, just as before (see Figure 1.4.4(a)).

In Figure 1.4.4(b) we see in which quadrants or on which axes the terminal side of an angle 0∘≤θ<360∘ may fall. From Figure 1.4.3(a) and formulas Equation ??? and Equation ???, we see that we can get negative values for a trigonometric function. For example, sinθ<0 when y<0. Figure 1.4.5 summarizes the signs (positive or negative) for the trigonometric functions based on the angle's quadrant:

Example 1.20
Find the exact values of all six trigonometric functions of 120∘.
Solution
We know 120∘=180∘−60∘. By Example 1.7 in Section 1.2, we see that we can use the point (−1,√3) on the terminal side of the angle 120∘ in QII, since we saw in that example that a basic right triangle with a 60∘ angle has adjacent side of length 1, opposite side of length √3, and hypotenuse of length 2, as in the figure on the right. Drawing that triangle in QII so that the hypotenuse is on the terminal side of 120∘ makes r=2, x=−1, and y=√3. Hence:
sin120∘=yr=√32cos120∘=xr=−12tan120∘=yx=√3−1=−√3
csc120∘=ry=2√3sec120∘=rx=2−1=−2cot120∘=xy=−1√3
Example 1.21
Find the exact values of all six trigonometric functions of 225∘.
Solution
We know that 225∘=180∘+45∘. By Example 1.6 in Section 1.2, we see that we can use the point (−1,−1) on the terminal side of the angle 225∘ in QIII, since we saw in that example that a basic right triangle with a 45∘ angle has adjacent side of length 1, opposite side of length 1, and hypotenuse of length √2, as in the figure on the right. Drawing that triangle in QIII so that the hypotenuse is on the terminal side of 225∘ makes r=√2, x=−1, and y=−1. Hence:
sin225∘=yr=−1√2cos225∘=xr=−1√2tan225∘=yx=−1−1=1
csc225∘=ry=−√2sec225∘=rx=−√2cot225∘=xy=−1−1=1
Example 1.22
Find the exact values of all six trigonometric functions of 330∘.
Solution
We know that 330∘=360∘−30∘. By Example 1.7 in Section 1.2, we see that we can use the point (√3,−1) on the terminal side of the angle 225∘ in QIV, since we saw in that example that a basic right triangle with a 30∘ angle has adjacent side of length √3, opposite side of length 1, and hypotenuse of length 2, as in the figure on the right. Drawing that triangle in QIV so that the hypotenuse is on the terminal side of 330∘ makes r=2, x=√3, and y=−1. Hence:
sin330∘=yr=−12cos330∘=xr=√32tan330∘=yx=−1√3
csc330∘=ry=−2sec330∘=rx=2√3cot330∘=xy=−√3
Example 1.23
Find the exact values of all six trigonometric functions of 0∘, 90∘, 180∘, and 270∘.

Solution
These angles are different from the angles we have considered so far, in that the terminal sides lie along either the x-axis or the y-axis. So unlike the previous examples, we do not have any right triangles to draw. However, the values of the trigonometric functions are easy to calculate by picking the simplest points on their terminal sides and then using the definitions in formulas Equation ??? and Equation ???.
For instance, for the angle 0∘ use the point (1,0) on its terminal side (the positive x-axis), as in Figure 1.4.6. You could think of the line segment from the origin to the point (1,0) as sort of a degenerate right triangle whose height is 0 and whose hypotenuse and base have the same length 1. Regardless, in the formulas we would use r=1, x=1, and y=0. Hence:
sin0∘=yr=01=0cos0∘=xr=11=1tan0∘=yx=01=0
csc0∘=ry=10=undefinedsec0∘=rx=11=1cot0∘=xy=10=undefined
Note that csc0∘ and cot0∘ are undefined, since division by 0 is not allowed.
Similarly, from Figure 1.4.6 we see that for 90∘ the terminal side is the positive y-axis, so use the point (0,1). Again, you could think of the line segment from the origin to (0,1) as a degenerate right triangle whose base has length 0 and whose height equals the length of the hypotenuse. We have r=1, x=0, and y=1, and hence:
sin90∘=yr=11=1cos90∘=xr=01=0tan90∘=yx=10=undefined
csc90∘=ry=11=1sec90∘=rx=10=undefinedcot90∘=xy=01=0
Likewise, for 180∘ use the point (−1,0) so that r=1, x=−1, and y=0. Hence:
sin180∘=yr=01=0cos180∘=xr=−11=−1tan180∘=yx=0−1=0
csc180∘=ry=10=undefinedsec180∘=rx=1−1=−1cot180∘=xy=−10=undefined
Lastly, for 270∘ use the point (0,−1) so that r=1, x=0, and y=−1. Hence:
sin270∘=yr=−11=−1cos270∘=xr=01=0tan270∘=yx=−10=undefined
csc270∘=ry=1−1=−1sec270∘=rx=10=undefinedcot270∘=xy=0−1=0
The following table summarizes the values of the trigonometric functions of angles between 0∘ and 360∘ which are integer multiples of 30∘ or 45∘:
Table 1.3 Table of trigonometric function values
Since 360∘ represents one full revolution, the trigonometric function values repeat every 360∘. For example, sin360∘=sin0∘, cos390∘=cos30∘, tan540∘=tan180∘, sin(−45∘)=sin315∘, etc. In general, if two angles differ by an integer multiple of 360∘ then each trigonometric function will have equal values at both angles. Angles such as these, which have the same initial and terminal sides, are called coterminal.
In Examples 1.20-1.22, we saw how the values of trigonometric functions of an angle θ larger than 90∘ were found by using a certain acute angle as part of a right triangle. That acute angle has a special name: if θ is a nonacute angle then we say that the reference angle for θ is the acute angle formed by the terminal side of θ and either the positive or negative x-axis. So in Example 1.20, we see that 60∘ is the reference angle for the nonacute angle θ=120∘; in Example 1.21, 45∘ is the reference angle for θ=225∘; and in Example 1.22, 30∘ is the reference angle for θ=330∘.
Example 1.24

- Which angle between 0∘ and 360∘ has the same terminal side (and hence the same trigonometric function values) as θ?
- What is the reference angle for θ?
Solution
(a) Since 928∘=2×360∘+208∘, then θ has the same terminal side as 208∘, as in Figure 1.4.7.
(b) 928∘ and 208∘ have the same terminal side in QIII, so the reference angle for θ=928∘ is 208∘−180∘=28∘.
Example 1.25
Suppose that cosθ=−45. Find the exact values of sinθ and tanθ.
Solution
We can use a method similar to the one used to solve Example 1.8 in Section 1.2. That is, draw a right triangle and interpret cosθ as the ratio adjacenthypotenuse of two of its sides. Since cosθ=−45, we can use 4 as the length of the adjacent side and 5 as the length of the hypotenuse. By the Pythagorean Theorem, the length of the opposite side must then be 3. Since cosθ is negative, we know from Figure 1.4.5 that θ must be in either QII or QIII. Thus, we have two possibilities, as shown in Figure 1.4.8 below:

When \theta is in QII, we see from Figure 1.4.8(a) that the point (-4,3) is on the terminal side of \theta , and so we have x = -4 , y = 3 , and r = 5 . Thus, \sin\;\theta = \frac{y}{r} = \frac{3}{5} and \tan\;\theta = \frac{y}{x} = \frac{3}{-4} .
When \theta is in QIII, we see from Figure 1.4.8(b) that the point (-4,-3) is on the terminal side of \theta , and so we have x = -4 , y = -3 , and r = 5 . Thus, \sin\;\theta = \frac{y}{r} = \frac{-3}{5} and \tan\;\theta = \frac{y}{x} = \frac{-3}{-4} = \frac{3}{4} .
Thus, either \fbox{\(\sin\;\theta = \frac{3}{5} \) and \(\tan\;\theta = -\frac{3}{4}\)} or \fbox{\(\sin\;\theta = -\frac{3}{5} \) and \(\tan\;\theta = \frac{3}{4}\)}.
Since reciprocals have the same sign, \csc\;\theta and \sin\;\theta have the same sign, \sec\;\theta and \cos\;\theta have the same sign, and \cot\;\theta and \tan\;\theta have the same sign. So it suffices to remember the signs of \sin\;\theta , \cos\;\theta , and \tan\;\theta:
For an angle θ in standard position and a point (x, y) on its terminal side:
- \sin\;\theta has the same sign as y
- \cos\;\theta has the same sign as x
- \tan\;\theta is positive when x and y have the same sign
- \tan\;\theta is negative when x and y have opposite signs
Contributors
Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2.