1. Oblique triangles

a. Formulae

An oblique triangle is one in which individual angles are greater than 0° and less than 180°. A right triangle is a special oblique triangle in which one angle is exactly 90°.

A plane triangle has six parts: three angles and three sides.

img1
 Figure B-1
Triangle Parts

  

Equations for triangle trigonometry are:

Angle Condition img3 Eqn (B-1)
Law of Sines img4 Eqn (B-2)
Law of Cosines img5 Eqn (B-3)

 

 A right triangle is a special case where one angle is exactly 90°. If we apply the Law of Cosines, Eqn (B-3), to a right triangle:

img6 

img8

 

 

 

 

 

The Law of Cosines becomes the Pythagorean Theorem.

Figure B-2
Right triangle
 

 

b. Solving triangles

In order to geometrically fix a triangle you must start with at least three parts and one of those must be a side. Why a side? Because fixing the angles alone does not constrain the size of the triangle.

img2  These two triangles have identical angles but are different sizes.
Figure B-2
Angles Don't Fix Triangle
 

 

 

Starting with three parts of a triangle, there is only one way to begin solving for another of its parts. Once a fourth part is determined, there are more ways to solve the remaing parts.

 

For example:

b ex1a Given three sides of a triangle, how would you begin solving for other parts of it?
b ex1b

Start with the Law of Cosines to solve for one angle.

a2=b2+c2-2x(bc)xCos(A)

Can then use Law of Cosines or Law of Sines to solve for next angle.

Can then use Law of Cosines or Law of Sines to solve for next angle.

Last angle can be solved by Law of Cosines, Law of Sines, or Angle Condition

 

b ex2a How about two sides and an included angle?
b ex2b

Start with the Law of Cosines to solve for side a (known values in red):

a2=b2+c2-2x(bc)xCos(A)

Can then use Law of Cosines or Law of Sines to solve for another angle.

Last angle can be solved by Law of Cosines, Law of Sines, or Angle Condition

 

How would you solve these triangles?

b ex3a Two angles and the included side.
b ex4a

Two angles an a non-included side.

b ex5a Two sides and a non-included angle.

 

c. Area

Another attribute of a triangle is its area which can be determined by one of two equations:

Using two sides and an included angle.   img9 Eqn (B-4)
Using all three sides - Heron's Formula.   img10 Eqn (B-5)

 

Given two parts of a triangle and its area, you should be able to determine its other parts, right?

b ex6a Starting with two sides and an area...
b ex6b

Solve angle A using the area equation:

Area = 1/2(bc)xCos (A)

Then use Law of Cosines to solve side a

a2=b2+c2-2(bc)Cos(A)

Then.....

b ex7a How about starting with an angle, opposite side, and triangle area?

 

2. Trigonometric Functions

a. Sine; Law of Sines

The sine of any angle falls between -1.0 and +1.0. Refer to the sine curve plot in Figure B-4 to see how angles and their sines relate. This curve repeats itself every 360°;
so sin(-340°) = sin(20°) = sin(380°)...

img11 
Figure B-4

 

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°. 

For example:

Sin(101°) is 0.981627

Sin-1(0.866025) is 79°

We started with 101° and wound up with 79°.

Sly obersverse that we are, we notice that 101°+79° is 180°

Table B-1 show some other angles, their sine behavior, relationship between the beginnng and ending angles.

Table B-1
α x=sin (α) sin-1(x) relationship
30° +0.50000 +30° α
150° +0.50000 +30° 180°-α
210° -0.50000 -30° 180°-α
330° -0.50000 -30° 360°+α

 

All of the angles in the third colum are between -90° and +90°. This is shown graphically in Figure B-5.

img12
Figure B-5
ArcSin Angle Range

 

A triangle may have an angle which is greater than 90° but if you are solving that angle using the Law of Sines, the computed angle will be less than 90°. Although no single angle in a triangle may exceed 180° the Law of Sines can result in two angles both less than 180°.

For example sin-1(0.5) can be 30° or 150° - both meet the sine condition but only one meets the triangle condition. A calculator will always return 30° even though the “true” angle is 150°. How do you know which is correct? It depends on the triangle.

(1) Example

In triangle ABC, a = 12.4', b = 8.7', and B = 36°40'.

Compute the remaining angles and side.

From the Law of Sines

img13

There are two angles whose sine is +0.85112

A = 58°20' and A = (180°00' - 58°20') = 121°40', Figure B-6.

 

img14 
Figure B-6
Multiple Angles with Same Sine

 

That means there are two different triangle solutions, Figure B-7

 img17
Figure B-7
Two Possible Angle Values

 

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.

Isolating the two triangles and solving their remaining components:

For A = 58°20'

 img15
Figure B-8
First Solution

 

Angle condition

img18

Law of Sines

 img19

 For A = 121°40'

 img16
 Figure B-9
Second Solution

 

Angle condition

img20

Law of Sines

img21

(2) Example 2

Compute the angle B in the following triangle: 

 img22
Figure B-10
Example 2

 

The missing angle can be computed two ways:

Law of Sines

img23

Angle condition

img24

As a result there are two "correct" answers fot the missing angle:

sin(108°) = sin(72°) = 0.95105652

Ans since neither exceeds 180°, either can be an angle of a triangle,

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

 

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. Refer to the cosine curve plot in Figure B-11 to see 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°. 

img25 
Figure B-11

 

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°; for example:

Table B-2
α x=sin (α) sin-1(x)
60° +0.50000 +60°
300° +0.50000 +60°
120° -0.50000 +120°
-240° -0.50000 +120°

 

All of the angles in the third colum are between 0° and +180°. This is shown graphically in Figure B-12.

img26
Figure B-12
ArcCos Angle Range

 

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 1

Compute the value of the angle R in the triangle below: 

img27
Figure B-13
Example

 

From the Law of Cosines:

img28

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 is not sinusoidal as are the sine and cosine plots. And unlike the other two it repeats itself every 180°;

img30
Figure B-14

 

The tangent curve is asymptotic at 90°, 270°, 450°, etc. Asymptotic means the curve gets close to, but never reaches, those values.
so tan(90°) = tan(270°) = ... = infinity

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. Why is that?

Recall that tan(α) = sin(α) / cos(α)

At 90°, cos(90°) = 0 so you get division by 0, hence tan(90°) = ∞.

There is a Law of Tangents, but we don't generally use it to solve triangles since the Laws of Sines or Cosines can be used instead. For extra credit, find and memorize it.