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3. Trigonometric Functions

a. Sine; Law of sines

The sine of any angle falls between -1.0 and +1.0. Figure C-5 shows how angles and their sines relate. This curve repeats itself every 360°;  so sin(-340°) = sin(20°) = sin(380°)...

Figure C-5

 Taking the arcsine (a.k.a. inverse sine, sin-1) of a number between -1.0 and +1.0 on a calculator will always return an angle between -90° and +90°, Figure C-6.

Figure C-6
-90° to +90°

Table C-1 shows how angles outside ±90° relate through sin() and sin-1():

Table C-1
a x = sin (a) b = sin-1(x) relationship
30° +0.500000 +30° b= a
150° +0.500000 +30° a = 180°- b
210° -0.500000 -30° a = 180°- b
330° -0.500000 -30° a = 360° + b


For example sin-1(0.5) = 30° or 150° but it can only be one or the other for a particular triangle. A calculator will always return 30° even though the correct angle may be 150°. How do you know which is correct? It depends on the triangle.

To solve a triangle using the Law of Sines you must have either two angles and a side or two sides and an angle. Moreover, one known side must be opposite a known angle.

(1) Example 1

In triangle ABC, a = 12.4', b = 8.7', and B = 36°40'. Compute the remaining angles and side.

From the Law of Sines, Equation C-2:

There are two angles whose sine is +0.85112: A = 58°20' and A = (180°00' - 58°20') = 121°40', Figure C-7.

Figure C-7
sin-1(+0.85112)

This means there are two possible triangles, Figure C-8.

Figure C-8
Two possible triangles

Consider point C as the center of an arc of radius 8.7'.  This arc can intersect the remaining side at two points; A1 and A2, creating two different triangles.

Compute the remaining components for each triangle:

For A = 58°20', Figure C-9:

Figure C-9
Law of sines example 1 solution 1

From the angle condition:, Equation C-1:

Using Law of Sines:

For A = 121°40', Figure C-10:

Figure C-10
Law of sines example 1 solution 2

From the angle condition

Using Law of Sines

Which triangle is the correct solution? It depends on the situation. Notice that the problem statement provided numbers but not a sketch. Had a sketch been included, then the correct triangle could have been selected.

(2) Example 2

Compute angle B in the triangle of Figure C-11.

Figure C-11
Law of sines example 2

The missing angle can be computed two ways:

- By Law of Sines

- By angle condition

As a result there are two “correct” answers for the missing angle;

sin(108°) = sin(72°) = 0.95105652

But only 108° fits both the Law of Sines and the Angle condition.

We can also visually determine which angle is correct, providing we have a reasonably drawn sketch. In Figure C-11 angle B is larger than 90° so we would select 108°.

Be careful when using the Law of Sines to solve for an unknown angle – there could be two possible answers only one of which will fit the particular triangle.

b. Cosine; Law of cosines

The cosine of any angle falls between -1.0 and +1.0. Figure C-12 shows how angles and their cosines relate. This curve repeats itself every 360°;  so cos(-340°) = cos(20°) = cos(380°)...  The cosine curve is identical to the sine curve except its phase differs by 90°.

Figure C-12

 Taking the arccos (a.k.a. inverse cosine, cos-1) of a number between -1.0 and +1.0 on a calculator will always return an angle between 0° and 180°, Figure C-13.

Figure C-13
0° to 180°

Table C-2 shows how angles outside ±180° relate through cos() and cos-1():

Table C-2
a x = cos(a) b = cos-1(x)
60° +0.500000 +60°
120° -0.500000 +120°
300° +0.500000 +60°
-240° -0.500000 +120°


Using the Law of Cosines will not cause an ambiguous solution as does the Law of Sines since any single angle in a triangle cannot exceed 180°.

To solve a triangle using the Law of Cosines you must have either three sides, or, two sides and an angle.

(1) Example

Compute the value of the angle R in the triangle of Figure C-14.

Figure C-14
Law of cosines example

From the Law of Cosines:

The Law of Cosines returns only one legitimate value when solving triangles.

c. Tangent

Unlike sine and cosine, the tangent of any angle not limited to the range of -1.0 to +1.0. As a matter of fact, the tangent range is ±(infinity). You can see that the tangent function plot, Figure C-15, is not sinusoidal as are the sine and cosine plots. And unlike the other two it repeats itself every 180°;

Figure C-15

The tangent curve is asymptotic at 90°, 270°, 450°, etc. Asymptotic means the curve gets close to, but never reaches, those values. Recall that tangent is sine divided by cosine. At 90°, sine = 1.0, cosine = 0.0, so tangent = 1.0/0.0 (with an identical pattern every 180°). The result of dividing by zero is infinity, hence the asymptotic plot.

Try evaluating tan(90°) on your calculator; you’ll probably get an error statement of sorts. Then try tan(89.99999°); you should get a pretty big number.

There is a Law of Tangents, but we don't generally use it to solve triangles since the Laws of Sines or Cosines are usually sufficient .

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