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TRIGONOMETRY
Areas of focus:
- Degrees versus radians
- Trigonometric functions
- Trigonometric relations between complementary angles
- Pythagorean Theorem
- The fundamental relation between sine and cosine
- The unit circle and visualizing the trigonometric functions
- Inverse of trigonometric functions
- Law of sines
- Law of cosines
- Values of trigonometric at specific angles
- Trigonometric identities
- Curves of sine, cosine, and tangent
- Approximation for small angles: sine, cosine, and tangent
Degrees Versus Radians:
One revolution is 360o^ , and is also 2 radians. Thus, due to linear proportionality of the two scales, the conversion from x degrees to y radians is:
One radian is equal to 3.14159..., and is normally approximated by 3.14. The following table gives equivalent angles in degrees, radians, and revolutions.
Degrees Radians Revolutions
0 o^0
30 o
45 o
60 o
90 o
180 o
270 o
360 o^1
Trigonometric Functions:
The trigonometric functions are named sine, cosine, tangent, cotangent, secant, and cosecant. A trigonometric function has one argument that is an angle and will be denoted
" ". In writing the trigonometric functions one uses the abbreviated forms: ,
, , , , and , respectively. Also, sometimes these are
written as , , , , , and , respectively.
The value of each trigonometric function for an acute angle (<90o^ ) can be directly related
to the sides of a right triangle. Consider the angle in the following figure. The values of the trigonometric functions for this angle are given as:
Since , we can also write:
Pythagorean Theorem:
The Pythagorean theorem states that for a right triangle, as shown, there exists a relation between the length of the sides given be
a 2 + b 2 = c 2
There are also Pythagorean triples for (a,b,c), such as (3,4,5), (5,12,13) and (7,24,25) sided triangles, and all constant multiples of these triplets (e.g., (6,8,10)).
Fundamental Relations Among Trigonometric Functions:
From the Pythagorean Theorem of plane geometry we know that x 2 + y^2 = r 2. This can be used to derive a basic relation between the sine and cosine functions.
GO TO PART 2
TRIGONOMETRY FOR STATICS
PART 2:
The Unit Circle and Visualizing Trigonometric Functions:
The fundamental relation suggests that the sine and cosine can be visualized by using a circle of unit radius. To do this, draw a circle of unit radius, as shown in the figure. Next draw a radial line from the center of the circle to the its arc and
making a counter clockwise angle with the horizontal axis as shown in the figure. The
projection of this line onto the horizontal axis is , the projection of this line onto
the vertical axis is , and if the radial line is extended to intersect the vertical line
AB one can get as shown in the figure.
At
At
At
Inverse of Trigonometric Functions:
The inverse trigonometric functions are: arcsine, arccosine, and arctangent. For a
specific value z , these are written as: , ,. For example, the
function provides the angles that has. In a similar manner,
and , respectively, provide the angles for which and .
For example, means the angle for which the sine has a value of 0.5.
Thus, one solution is. Likewise, has a solution.
The inverse trigonometric functions are also written as sin-1^ , cos -1^ , and tan -1^. For example,
is the same as. This contradicts the convention established for
positive exponents. Therefore, even though
Law of Sines:
The law of sines states that
This can be shown by considering the triangles AXB and CXB in the following figure.
We have and , hence or.
In a similar manner one can show that.
Law of Cosines:
The law of cosines states that
This can be shown by considering the triangle BXC that gives:
a 2 = p 2 + (CX ) 2 = p 2 + (b - AX)^2
or a 2 = p^2 + b 2 + (AX) 2 - 2b(AX) (1)
Considering the triangle AXB one gets:
p^2 + (AX) 2 = c 2 and
Substituting these into (1) one obtains:
The other relations are obtained in a similar manner.
GO TO PART 3
GO BACK TO PART 1
TRIGONOMETRY FOR STATICS
PART 3:
Trigonometric Identities:
Basic identities:
Half-angles:
Identities in terms of tan (/2):
where
Curves of Sine, Cosine, and Tangent:
Sine function:
The sine function is an odd function since
replace with
replace with
replace with unity (i.e., )
