trigonometry

1. TRIGONOMETRY
http://math.la.asu.edu/~tdalesan/mat170/TRIGONOMETRY.ppt

2. Angles, Arc length, Conversions
Angle measured in standard position.Angle measured in standard position.
Initial side is the positive x – axis which is fixed.Initial side is the positive x – axis which is fixed.
Terminal side is the ray in quadrant II, which is freeTerminal side is the ray in quadrant II, which is free
to rotate about the origin. Counterclockwise rotationto rotate about the origin. Counterclockwise rotation
is positive, clockwise rotation is negative.is positive, clockwise rotation is negative.
Coterminal Angles: Angles that have the same terminal side.Coterminal Angles: Angles that have the same terminal side.
60°, 420°, and –300° are all coterminal.60°, 420°, and –300° are all coterminal.
Degrees to radians: Multiply angle byDegrees to radians: Multiply angle by .
180
π
3180
60
ππ
=× 

radiansradians
Radians to degrees: Multiply angle byRadians to degrees: Multiply angle by .
180
π



45
180
4
=×
π
π
Arc length = central angle x radius, orArc length = central angle x radius, or .rs θ=
Note: The central angle must be in radian measure.Note: The central angle must be in radian measure.
Note: 1 revolution = 360° = 2π radians.Note: 1 revolution = 360° = 2π radians.

3. Right Triangle Trig Definitions
• sin(A) = sine of A = opposite / hypotenuse = a/c
• cos(A) = cosine of A = adjacent / hypotenuse = b/c
• tan(A) = tangent of A = opposite / adjacent = a/b
• csc(A) = cosecant of A = hypotenuse / opposite = c/a
• sec(A) = secant of A = hypotenuse / adjacent = c/b
• cot(A) = cotangent of A = adjacent / opposite = b/a
A
a
b
c
B
C

4. Special Right Triangles
30°30°
45°
60° 45°
2
11
3 1
1
2
3
3
)30tan(
2
1
)30sin(
2
3
)30cos(
=
=
=



3)60tan(
2
3
)60sin(
2
1
)60cos(
=
=
=



1)45tan(
2
2
)45sin(
2
2
)45cos(
=
=
=




5. Basic Trigonometric Identities
)cos(
)sin(
)tan(
A
A
A =
)sin(
)cos(
)cot(
A
A
A =
)csc(
1
)sin(
)sin(
1
)csc(
A
A
A
A
=
=
)sec(
1
)cos(
)cos(
1
)sec(
A
A
A
A
=
=
)cot(
1
)tan(
)tan(
1
)cot(
A
A
A
A
=
=
1)(cos)(sin 22
=+ AA
)(sec1)(tan 22
AA =+ )(csc)(cot1 22
AA =+
Quotient identities:
Reciprocal Identities:
Pythagorean Identities:
Even/Odd identities:
)csc()csc(
)sin()sin(
AA
AA
−=−
−=−
)cot()cot(
)tan()tan(
AA
AA
−=−
−=−
)sec()sec(
)cos()cos(
AA
AA
=−
=−
Even functions Odd functions Odd functions

6. AAll SStudents TTake CCalculus.
Quad II
Quad I
Quad III Quad IV
cos(A)>0
sin(A)>0
tan(A)>0
sec(A)>0
csc(A)>0
cot(A)>0
cos(A)<0
sin(A)>0
tan(A)<0
sec(A)<0
csc(A)>0
cot(A)<0
cos(A)<0
sin(A)<0
tan(A)>0
sec(A)<0
csc(A)<0
cot(A)>0
cos(A)>0
sin(A)<0
tan(A)<0
sec(A)>0
csc(A)<0
cot(A)<0

7. Reference Angles
Quad IQuad I
Quad IIQuad II
Quad IIIQuad III Quad IVQuad IV
θ’ = θθ’ = 180° – θ
θ’ = θ – 180° θ’ = 360° – θ
θ’ = π – θ
θ’ = 2π – θθ’ = θ – π

8. Unit circle
• Radius of the circle is 1.
• x = cos(θ)
• y = sin(θ)
• Pythagorean Theorem:
• This gives the identity:
• Zeros of sin(θ) are where n is an integer.
• Zeros of cos(θ) are where n is an
integer.
1)sin(1 ≤≤− θ
1)cos(1 ≤≤− θ
122
=+ yx
1)(sin)(cos 22
=+ θθ
πn
π
π
n+
2

10. Graphs of sine & cosine
• Fundamental period of sine and cosine is 2π.
• Domain of sine and cosine is
• Range of sine and cosine is [–|A|+D, |A|+D].
• The amplitude of a sine and cosine graph is |A|.
• The vertical shift or average value of sine and
cosine graph is D.
• The period of sine and cosine graph is
• The phase shift or horizontal shift is
DCBxAxg
DCBxAxf
+−=
+−=
)cos()(
)sin()(
.
2
B
π
.
B
C
.ℜ

11. Sine graphs
y = sin(x)
y = sin(3x)
y = 3sin(x)
y = sin(x – 3)
y = sin(x) + 3
y = 3sin(3x-9)+3
y = sin(x)
y = sin(x/3)

12. Graphs of cosine
y = cos(x)
y = cos(3x)
y = cos(x – 3)
y = 3cos(x)
y = cos(x) + 3
y = 3cos(3x – 9) + 3
y = cos(x)
y = cos(x/3)

13. Tangent and cotangent graphs
• Fundamental period of tangent and cotangent is
π.
• Domain of tangent is where n is an
integer.
• Domain of cotangent where n is an
integer.
• Range of tangent and cotangent is
• The period of tangent or cotangent graph is
DCBxAxg
DCBxAxf
+−=
+−=
)cot()(
)tan()(






+≠ π
π
nxx
2
|
{ }πnxx ≠|
.ℜ
.
B
π

14. Graphs of tangent and cotangent
y = tan(x)
Vertical asymptotes at
y = cot(x)
Verrical asymptotes at .πnx =.
2
π
π
nx +=

15. Graphs of secant and cosecant
y = sec(x)
Vertical asymptotes at
Range: (–∞, –1] U [1, ∞)
y = cos(x)
y = csc(x)
Vertical asymptotes at
Range: (–∞, –1] U [1, ∞)
y = sin(x)
.
2
π
π
nx += .πnx =

16. Inverse Trigonometric Functions
and Trig Equations
)arctan()(tan 1
xxy == −
)arcsin()(sin 1
xxy == −
)arccos()(cos 1
xxy == −




−
2
,
2
ππ
Domain: [–1, 1]
Range:
0 < y < 1, solutions in QI and QII.
–1 < y < 0, solutions in QIII and QIV.
Domain: [–1, 1]
Range: [0, π]
0 < y < 1, solutions in QI and QIV.
–1< y < 0, solutions in QII and QIII.






−
2
,
2
ππ
Domain:
Range:
0 < y < 1, solutions in QI and QIII.
–1 < y < 0, solutions in QII and QIV.
ℜ

17. Trigonometric Identities
Summation & Difference Formulas
)tan()tan(1
)tan()tan(
)tan(
)sin()sin()cos()cos()cos(
)sin()cos()cos()sin()sin(
BA
BA
BA
BABABA
BABABA


±
=±
=±
±=±

18. Trigonometric Identities
Double Angle Formulas
)(tan1
)tan(2
)2tan(
1)(cos2)(sin21)(sin)(cos)2cos(
)cos()sin(2)2sin(
2
2222
A
A
A
AAAAA
AAA
−
=
−=−=−=
=

19. Trigonometric Identities
Half Angle Formulas
)cos(1
)cos(1
2
tan
2
)cos(1
2
cos
2
)cos(1
2
sin
A
AA
AA
AA
+
−
±=





+
±=





−
±=




 The quadrant of 2
A
determines the sign.

20. Law of Sines & Law of Cosines
)sin()sin()sin(
)sin()sin()sin(
C
c
B
b
A
a
c
C
b
B
a
A
==
==
)cos(2
)cos(2
)cos(2
222
222
222
Abccba
Baccab
Cabbac
−+=
−+=
−+=
Law of sines Law of cosines
Use when you have a
complete ratio: SSA.
Use when you have SAS, SSS.

21. Vectors
• A vector is an object that has a magnitude and a direction.
• Given two points P1: and P2: on the plane, a
vector v that connects the points from P1 to P2 is
v = i + j.
• Unit vectors are vectors of length 1.
• i is the unit vector in the x direction.
• j is the unit vector in the y direction.
• A unit vector in the direction of v is v/||v||
• A vector v can be represented in component form
by v = vxi + vyj.
• The magnitude of v is ||v|| =
• Using the angle that the vector makes with x-axis in
standard position and the vector’s magnitude, component
form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j
22
yx vv +
),( 11 yx ),( 22 yx
)( 12 xx − )( 12 yy −

22. Vector Operations
Scalar multiplication: A vector can be multiplied by any scalar (or number).
Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j.
Dot Product: Multiplication of two
vectors.
Let v = vxi + vyj, w = wxi + wyj.
v · w = vxwx + vywy
Example: Let v = 5i + 4j, w = –2i + 3j.
v · w = (5)(–2) + (4)(3) = –10 + 12 = 2.
Two vectors v and w are orthogonal (perpendicular) iff v · w = 0.
Addition/subtraction of vectors: Add/subtract same components.
Example Let v = 5i + 4j, w = –2i + 3j.
v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j.
3v – 2w = 3(5i + 4j) – 2(–2i + 3j) = (15i + 12j) + (4i – 6j) = 19i + 6j.
||3v – 2w|| = 9.19397619 22
≈=+
Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the
angle between the two vectors.
θ
w
v

23. Acknowledgements
• Unit Circle: http://www.davidhardison.com/math/trig/unit_circle.gif
• Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.