Units outside the SI that are accepted for use with
1.1.7 Sine/Cosine Law’s
The Six Basic Trigonometric Functions Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle.
1.1 Preparatory Concepts | 31
To define the trigonometric functions, first consider the unit circle centered at the origin and a
point P=(x,y)P=(x,y)" role="presentation" style="overflow:
initial;font-style: normal;font-weight: normal;line- height: normal;font-size: 14px;text-indent: 0px;text- align: left;text-transform: none;letter-spacing:
normal;float: none;direction: ltr;max-width: none;max- height: none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">P=(x,y) on the unit circle. Let θθθ" role="presentation" style="overflow:
initial;font-style: normal;font-weight: normal;line- height: normal;font-size: 14px;text-indent: 0px;text- align: left;text-transform: none;letter-spacing:
normal;float: none;direction: ltr;max-width: none;max- height: none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px"> be an angle with an initial side that lies along the positive xxx"
role="presentation" style="overflow: initial;font-style:
normal;font-weight: normal;line-height: normal;font- size: 14px;text-indent: 0px;text-align: left;text- transform: none;letter-spacing: normal;float:
none;direction: ltr;max-width: none;max-height:
none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">-axis and with a terminal side that is the line segment OP. We can then define the values of the six trigonometric functions for θ θθ"
role="presentation" style="overflow: initial;font-style:
normal;font-weight: normal;line-height: normal;font- size: 14px;text-indent: 0px;text-align: left;text- transform: none;letter-spacing: normal;float:
none;direction: ltr;max-width: none;max-height:
none;min-width: 0px;min-height: 0px;border:
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0px;padding: 0px;margin: 0px">in terms of the coordinates x xx" role="presentation" style="overflow:
initial;font-style: normal;font-weight: normal;line- height: normal;font-size: 14px;text-indent: 0px;text- align: left;text-transform: none;letter-spacing:
normal;float: none;direction: ltr;max-width: none;max- height: none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">and y.y.y."
role="presentation" style="overflow: initial;font-style:
normal;font-weight: normal;line-height: normal;font- size: 14px;text-indent: 0px;text-align: left;text- transform: none;letter-spacing: normal;float:
none;direction: ltr;max-width: none;max-height:
none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">
Let P=(x,y) be a point on the unit circle centered at the origin O. Let θθ" role="presentation" style="overflow:
1.1 Preparatory Concepts | 33
initial;font-style: normal;font-weight: normal;line- height: normal;font-size: 14px;text-indent: 0px;text- align: left;text-transform: none;letter-spacing:
normal;float: none;direction: ltr;max-width: none;max- height: none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">θ be an angle with an initial side along the positive x-axis and a terminal side given by the line segment OP. The trigonometric functions are then defined as
$$\sin\theta=y\;\;\;\csc\theta=\frac{1}{y}\\\cos\
theta=x\;\;\;\sec\theta=\frac{1}{x}\\\tan\
theta=\frac{y}{x}\;\;\;\cot\theta=\frac{x}{y}$$
If x=0x=0,secθx=0,secθ" role="presentation"
style="overflow: initial;font-style: normal;font-weight:
normal;line-height: normal;font-size: 14px;text-indent:
0px;text-align: left;text-transform: none;letter-spacing:
normal;float: none;direction: ltr;max-width: none;max- height: none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">, secθ and tanθ are undefined. If y=0, then cotθ and cscθ are undefined.
We can see that for a point P=(x,y) on a circle of radius r with a corresponding angle θ,θ," role="presentation"
style="overflow: initial;font-style: normal;font-weight:
normal;line-height: normal;font-size: 14px;text-indent:
34 | Statics
0px;text-align: left;text-transform: none;letter-spacing:
normal;float: none;direction: ltr;max-width: none;max- height: none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">θ, the coordinates x and y satisfy:
[latex]\cos \theta =\frac {x}{r}[/latex]
[latex]x=r\cos\theta[/latex]
[latex]\sin \theta =\frac {y}{r}[/latex]
[latex]x=r\sin\theta[/latex]
The values of the other trigonometric functions can be expressed in terms of x, y, and r:
The table below shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values
1.1 Preparatory Concepts | 35
of the other trigonometric functions are calculated easily from the values of sinθ and cosθ:
Trigonometric Identities
A trigonometric identity is an equation involving trigonometric functions that is true for all angles θθ"
role="presentation" style="overflow: initial;font-style:
normal;font-weight: normal;line-height: normal;font- size: 14px;text-indent: 0px;text-align: left;text- transform: none;letter-spacing: normal;float:
none;direction: ltr;max-width: none;max-height:
none;min-width: 0px;min-height: 0px;border:
0px;padding: 0px;margin: 0px">θ for which the
functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next.
36 | Statics
Source: Calculus Volume 1, Gilbert Strang & Edwin
“Jed” Herman, https://openstax.org/books/calculus- volume-1/pages/1-3-trigonometric-functions
We often refer to this as SOH-CAH-TOA:
• Sine = Opposite / Hypotenuse >> S = O/H >> SOH
• Cosine = Adjacent / Hypotenuse >> C = A / H >> CAH
• Tangent = Opposite / Adjacent >> T = O / A >> TOA
1.1 Preparatory Concepts | 37
I remember that cos is close – the side that’s close to the angle is cosine. (It kind of rhymes and ‘close’ is a more familiar word than
‘adjacent’).
Key Takeaways
Basically: Trigonometric functions will help you to solve problems. You’ll use SOH-CAH-TOA in many statics problems, whether to componentize a vector or resolve a force.
Application: A 6ft ladder leaning up against a house is at a 60 degree angle. We can find the vertical height where the ladder reaches the house by using height = 6 ft * sin 60 degrees. (sin = opp / hyp)
Looking Ahead: Chapter 4 (forces) and Chapter 5 (trusses) will use calculation of angles a lot.
38 | Statics