Download Trigonometry Notes, Definitions, and Formulae

Trigonometry Notes, Definitions, and Formulae
Definitions
•
Degree – 1/360 of a complete revolution.
•
Minute – 1/60 of a degree
•
Second – 1/60 of a minute or (1/60 · 1/60) of a degree
•
Initial side – the beginning ray of an angle
•
Terminal side – the ray that moves to form an angle with the initial side
•
Directed angle – angle that can be described as positive or negative
•
Standard position – the position of an angle whose initial side coincides with the positive x – axis
•
Quadrantal angle – angle whose terminal side lies on a coordinate axis
•
Coterminal – used to describe two angles whose terminal sides coincide
•
Trigonometric identity – an equation involving trigonometric functions of an angle θ that is true for
all values of θ (e.g. tan θ = sin θ/cos θ)
•
Pythagorean identities – sin² θ + cos² θ = 1; tan² θ + 1 = sec² θ; cot² θ + 1 = csc² θ
•
Cofunctions – any pair of trigonometric functions of complementary acute angles that are equal (e.g.
– in a triangle with complementary angles A and B, sin A = cos B)
•
Distance formula – r = x² + y²
•
Reference angle of θ – a unique acute angle α, corresponding to θ, formed by the terminal side of θ
and the positive or negative x – axis
•
Reference triangle – triangle drawn using the reference angle α to be used to find values for
trigonometric functions of θ
•
Solving the triangle – finding the measurements of all three sides and all three angles
•
Angle of elevation and angle of depression – angles ascending or descending from horizontal for
application in word problems; they equal each other
•
Radian – the ratio of the length of the curve or arc to the radius: θ = _s_ and s = rθ
r
•
Circular functions – functions that use a circle in their definition but that are used for nongeometric
applications like ocean waves or alternating current
•
Periodic – repeating. A function is periodic if for some positive constant p f(x+p) = f(x) for every x
in the domain of f. The smallest such p is the period of f.
•
Odd function – in general f is an odd function if f(–x) = –(f)x When f is odd, the graph is symmetric
with respect to the origin. If the point (x, y) is on the graph of f, then so is the point (–x, –y).
•
Even function – in general f is an even function if f(–x) = f(x) When f is even, the graph is symmetric
with respect to the y-axis. If the point (x, y) is on the graph of f, then so is the point (–x, y).
•
Amplitude – designated by the letter a it is the difference between the minimum and maximum
values of a curve divided by 2. M – m = (c + a) – (c – a) = 2c = a
2
2
2
•
Sine and cosine functions are expressed by the formulas y = c + a sin bx and y = c + a cos bx In
these expressions, a is amplitude and c is the number of units the function is shifted above or below
the standard position. M + m = (c + a) + (c – a) = 2c = c The coefficient b affects the period
2
2
2
because the functions have a period of 2π
b
Trigonometric Functions of Acute Angles
sin θ = _y_ =
r
cos θ = _x_ =
r
tan θ = _y_ =
x
__opposite__
hypotenuse
__adjacent__
hypotenuse
__opposite__
adjacent
Reciprocal functions of acute angles
cot θ = _x_ = __adjacent__ = _1_
y
opposite
tan θ
sec θ = _r_ = __hypotenuse__ = _1_
x
adjacent
cos θ
csc θ = _r_ = __hypotenuse__ = _1_
y
opposite
sin θ
Cofunctions of complementary angles
sin A = cos B
cos A = sin B
tan A = cot B
cot A = tan B
sec A = csc B
csc A = sec B
Trigonometric functions for 30º, 45º, and 60º angles
θ
30º
45º
60º
sin θ
_1_
2
2
2
3
2
cos θ
3
2
2
2
_1_
2
tan θ
3
3
csc θ
1
2
2 3
3
3
2
sec θ
2 3
3
cot θ
2
1
2
3
3
3
Trigonometric function values for general triangles
Function
Quadrant of θ
value
I
II
III
IV
sin θ
csc θ
+
+
–
–
cos θ
sec θ
+
–
–
+
tan θ
cot θ
+
–
+
–
Law of Cosines – In any triangle ABC,
c² = a² + b² – 2ab cos C
b² = a² + c² – 2ac cos B
a² = b² + c² – 2bc cos A
cos C = a² + b² – c²
2ab
Law of Sines – In any triangle ABC,
sin A = sin B = sin C
and also a = sin A
a
b
c
b
sin B
Circular Functions
sin s = y
tan s = sin s if cos s ≠ 0
cos s
sec s = _1_ if cos s ≠ 0
cos s
cos s = x
cot s = cos s if sin s ≠ 0
sin s
csc s = _1_ if sin s ≠ 0
sin s
Degree and radian measures of some frequently used angles
Degree
measure
0º
30º
45º
60º
90º
120º
135º
Radian
measure
0
_π_
6
_π_
4
_π_
3
_π_
2
_2π_
3
_3π_
4
150º
180º
_5π_
6
π
The Reciprocal Identities
sin α = _1_
csc α
sin α csc α = 1
csc α = _1_
sin α
cos α = _1_
sec α
cos α sec α = 1
sec α = _1_
cos α
_1_
cot α
tan α =
_1_
tan α
tan α cot α = 1
cot α =
sin α
cos α
The Cofunction Identities
sin θ = cos (90º – θ )
tan θ = cot (90º – θ )
sec θ = csc (90º – θ )
The Pythagorean Identities
sin² α + cos² α = 1
1 + tan² α = sec² α
cos α
sin α
cos θ = sin (90º – θ )
cot θ = tan (90º – θ )
csc θ = sec (90º – θ )
1 + cot² α = csc² α
Notes:
Counterclockwise rotations of the terminal side produce positive angles.
Clockwise rotations of the terminal side produce negative angles.
Angles can be classified according to the quadrant in which their terminal side lies.
Use the MODE button on your calculator to switch from degrees to radians or vice versa.
If θ is a complete revolution, s = 2πr and θ = 2πr = 2π
180 = π radians
R
1º = _π_ radians
180
1 radian = 180º ≈ 57.3º
π
The product of two odd functions is even.
The product of an even function and an odd function is odd.
The sum of an even function and an odd function is neither.
Sine and cosine functions have period 2π.
Sine is an odd function. Cosine is an even function. Use that information to derive whether each of the
other four functions are odd or even.
If a cosine curve were shifted π units to the right, it would coincide with the sine curve.
2
Tangent is an odd function that snakes back and forth between two asymptotes and has period π.
General Strategies for Proving Identities
1. Simplify the more complicated side of the identity until it is identical to the other side.
2. Transform both sides of the identity into the same expression.
Special Strategies for Proving Identities
1. Express functions in terms of sines and cosines.
2. Look for expressions to which the Pythagorean identities can be applied.
3. Use factoring.
4. Combine terms on each side of the identity into a single fraction.
5. Multiply one side of the equation by an expression equal to one.
Addition Formulas for the Sine and Cosine
sin (α + β) = sin α cos β + cos α sin β
sin (α – β) = sin α cos β – cos α sin β
cos (α + β) = cos α cos β – sin α sin β
cos (α – β) = cos α cos β + sin α sin β
Double-Angle Formulas for Sine and Cosine
sin 2 α = 2 sin α cos α
cos 2 α = cos² α – sin² α
cos 2 α = 1 – 2 sin² α
cos 2 α = 2 cos² α – 1
Half-Angle Formulas for Sine and Cosine
sin _θ_ = ± 1 – cos θ
cos _θ_ = ± 1 + cos θ
2
2
2
2
Addition Formulas for the Tangent
tan (α + β) = _tan α + tan β_
1 – tan α tan β
tan (α – β) = _tan α – tan β_
1 + tan α tan β
Double-Angle Formulas for the Tangent
tan 2 α = __2 tan α_
1 – tan² α
Half-angle Formulas for the Tangent
tan _θ_ = ± 1 – cos θ
tan _θ_ = ± __sin θ__
2
1 + cos θ
2
1 + cos θ
cot (α + β) = cot α cot β – 1
cot α + cot β
cot (α – β) = cot α cot β + 1
cot β – cot α
tan _θ_ = 1 – cos θ
2
sin θ