Angles and Their Measures

An angle is two rays with a common endpoint.

 To measure an angle, we draw a unit circle, centered around the common endpoint O of the rays of the angle. The measure of the angle in radians is then defined as the length of the arc of the circle between the points P and Q of intersection of the circle and the rays. The greek letters θ (theta) and φ (phi) are commonly used to denote angular measure. As one can see in the figure, the line segments OPand OQ each have length 1, while the arc PQ has length θ, which we then call the measure of the angle ∠POQ.

As an example,  consider a right triangle, as shown below. What is its measure in radians? Since the corresponding arc is one-quarter of the unit circle, whose total arc length is 2π, we see that the measure of a right angle is 2π/4 = π/2 radians.

Besides radians, angles are commonly measured in another unit called degrees. The measure of an angle in degrees is equal to 180/π times its measure in radians.Thus, the measure of a right angle is (180/π)(π/2) = 90 degrees, also written as 90°, and the measure of a full circle is (180/π)(2π) = 360 degrees, or 360°. To convert from degrees to radians, we multiply by π/180. Thus, for instance, an angle of 60° is equal to (60)(π/180) = π/3 radians.

Converting Degrees to Radians

1) 45°

2) 120°

3) 180°

4) 1°

Solution:

1) We have 45° = (45)(π/180) = π/4 radians.

2) We have 120° = (120)(π/180) = 2π/3 radians.

3) We have 180° = (180)(π/180) = π radians.

4) We have 1° = (1)(π/180) = π/180 radians.

Converting Radians to Degrees

1) π

2) π/6

3) 3π/4

4) 1

Solution:

1) We have π radians = (π)(180/π) = 180°.

2) We have π/6 radians = (π/6)(180/π) = 30°.

3) We have 3π/4 radians = (3π/4)(180/π) = 135°.

4) We have1 radian = (1)(180/π) = 180/π degrees ≈ 57.3°.

Angles in Quadrant

An angle in standard position is in the quadrant where its terminal side lies. Look at the first quadrant there is a 45° angle in standard position. In the II quadrant we have 135°, in the III we have 225° and last, in the IV we have 315°. As you can see in the figure below there are 2 arrows between the x-axis and the terminal side, which we will determine on the next topic.

Reference Angle

The reference angle for a nonquadrantal angle greater than 90° is the smallest nonnegative angle between the terminal side and the x axis when the angle is in standard position. It is drawn from the x axis to the terminal side of the angle and sometimes from the terminal side to the x axis but always in counter clockwise direction. All angles in standard position less than 90° are reference angles. You can look at the figure above for examples. Our reference angles for those examples are all 45°.

Sine, Cosine, and Tangent Functions

Values of Sin-Cos-Tan 

Sine, Cosine and Tangent are all based on a Right-Angled Triangle.

• "Opposite" is opposite to the angle θ
• "Adjacent" is adjacent (next to) to the angle θ
• "Hypotenuse" is the long one

Adjacent is always next to the angle

And Opposite is opposite the angle

For a triangle with an angle θ, they are calculated this way:

Sine Function:	sin(θ) = Opposite / Hypotenuse
Cosine Function:	cos(θ) = Adjacent / Hypotenuse
Tangent Function:	tan(θ) = Opposite / Adjacent

In picture form:

How to remember? Think "Sohcahtoa"! It works like this:

Soh...	Sine = Opposite / Hypotenuse
...cah...	Cosine = Adjacent / Hypotenuse
...toa	Tangent = Opposite / Adjacent

Graphs of Sin-Cos-Tan Functions


Plot of Sine

The Sine Function has this beautiful up-down curve (which repeats every 2πradians, or 360°).

It starts at 0, heads up to 1by π/2 radians (90°) and then heads down to -1.

Plot of Cosine

Cosine is just like Sine, but it starts at 1 and heads down until π radians (180°) and then heads up again.

Plot of Sine and Cosine

In fact Sine and Cosine are like good friends: they follow each other, exactly "π/2" radians, or 90°, apart.

Plot of the Tangent Function

The Tangent function has a completely different shape ... it goes between negative and positive Infinity, crossing through 0 (everyπ radians, or 180°), as shown on this plot.

At π/2 radians, or 90° (and -π/2, 3π/2, etc) the function is officiallyundefined, because it could be positive Infinity ornegative Infinity.

Properties of Sin-Cos-Tan Functions

The sine and cosine functions take on y-values

between -1 and 1. 

Sine Function:  y = sin x      called a "wave" because of its rolling wave-like           appearance (also referred to as oscillating)      amplitude: 1  (height)      period:  2π (length of one cycle)      frequency:  1 cycle in 2π radians [or 1/(2π)]      domain:        range:    At x = 0, the sine wave is on the shoreline! (meaning the y-value is equal to zero)

Cosine Function: y  = cos x       called a "wave" because of its rolling wave-like        appearance      amplitude: 1      period:  2π      frequency:  1 cycle in 2 radians [or 1/(2π)]      domain:        range:    At x = 0, the cosine wave breaks off the cliff! (meaning the y-value is equal to one) Tangent Function: y  =  tan x         domain: all real numbers except π/2 + k π, k is an integer.         range: all real numbers         period = π        x intercepts: x = kπ  , where k is an integer.        y intercepts: y = 0        symmetry: since tan(-x) = - tan(x) then tan (x) is an odd function and its graph is symmetric with respect the origin.         intervals of increase/decrease: over one period and from -π/2 to π/2, tan (x) is increasing.         vertical asymptotes: x = π/2 + k π, where k is an integer.

Trigonometric Identities

Magic Hexagon for Trig Identities

• Reciprocal Identities

You can also get the "Reciprocal Identities", by going "through the 1

Here you can see that sin(x) = 1 / csc(x)

sin(x) = 1 / csc(x) cos(x) = 1 / sec(x) cot(x) = 1 / tan(x) csc(x) = 1 / sin(x) sec(x) = 1 / cos(x) tan(x) = 1 / cot(x)

• Product Identities

The hexagon also shows that a function between any two functions is equal to them multiplied together (if they are opposite each other, then the "1" is between them):

Example: tan(x)cos(x) = sin(x)	Example: tan(x)cot(x) = 1

sin(x)csc(x) = 1
tan(x)csc(x) = sec(x)
sin(x)sec(x) = tan(x)
cos(x)tan(x) = sin(x)
csc(x)cos(x) = cot(x)
cot(x)sec(x) = csc(x)

• Quotient Identities 

tan(x) = sin(x) / cos(x) To help you remember think "tsc !"

Well, we can now follow "around the clock" (either direction) to get all the "Quotient Identities":

Clockwise

tan(x) = sin(x) / cos(x) sin(x) = cos(x) / cot(x) cos(x) = cot(x) / csc(x) cot(x) = csc(x) / sec(x) csc(x) = sec(x) / tan(x) sec(x) = tan(x) / sin(x)

Counterclockwise

cos(x) = sin(x) / tan(x) sin(x) = tan(x) / sec(x) tan(x) = sec(x) / csc(x) sec(x) = csc(x) / cot(x) csc(x) = cot(x) / cos(x) cot(x) = cos(x) / sin(x)

• Pythagorean Identities 

The Unit Circle shows us that

sin² x + cos² x = 1

The magic hexagon can help us remember that, too, by going clockwise around any of these three triangles:

And we have:

• sin²(x) + cos²(x) = 1
• 1 + cot²(x) = csc²(x)
• tan²(x) + 1 = sec²(x)