Algebra 2 Study Sheet pp
Pythagorean Identities
2
2
-
𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 = 1
-
1 + 𝑡𝑎𝑛 𝑥 = 𝑠𝑒𝑐 𝑥
-
1 + 𝑐𝑜𝑡 𝑥 = 𝑐𝑠𝑐 𝑥
2
2
2
2
Verifying Trigonometric Identities
-
Strategies for verifying trigonometric identities:
1. Convert to sine and cosine
2. A + B -> C (find a common denominator of A and B then add)
3. C -> A + B (split C into two different fractions)
𝑎
4. Turn division into multiplication ( 𝑏 = 𝑎 ×
1
5. Multiply by the conjugate ( 𝑥−𝑦 −>
1
𝑏
)
1(𝑥+𝑦)
)
(𝑥−𝑦)(𝑥+𝑦)
6. Factor the expression (𝑥𝑦 + 𝑥𝑧 = 𝑥(𝑦 + 𝑧))
Translations of Sine and Cosine Graphs
-
y = asin k(x − b) + c or y = acos k(x − b) + c
- Amplitude:The distance from the highest or lowest point of a trigonometric and
the midline.
- Midline: The median distance from the highest and lowest equations
- Phase Shift: The distance the equation has moved from 0,0 horizontally (if
negative, go to the right, if positive go to the left)
- Vertical Shift: The distance the equation has moved from 0,0 vertically ( if the
vertical shift is positive, go up. If the vertical shift is negative, go down/
- Vertical Stretch:
- Horizontal Compression:
Six Trigonometric Graphs
-
Cosine
Secant
Range:
𝑥 ≤− 1 ∨ 𝑥 ≥ 1
Domain:
(− ∞, − 1] ∨ [1, ∞)
-
Sine
Cosecant
Range:
𝑥 ≤− 1 ∨ 𝑥 ≥ 1
Domain:
(− ∞, − 1] ∨ [1, ∞)
-
Tangent
Cotangent
Range:
All real numbers
Domain:
all real numbers except for asymptotes
y = g cos d(x - r) + w
r = positive, right
- (x - 20) means 20 right
r = negative, left
- (x - - 20) or (x + 20) means 20 left
2pi / freq = period for cos/sec/sin/csc
pi / freq = period for tan/cot
Finding variables:
Period- to find the period of a graph you must divide the pi value of an equation by the
frequency.
● Cos pi value=2pi
● sin pi value=2pi
● Tan pi value= pi
In the equation of y=4/5sinx the period would be found by dividing 2pi/(⅘)
Phase shift- Phase shift is represented as value c above, when the phase shift is positive then
you move to the left, when the phase shift is negative you move to the right. For example the
equation y=sin(x-3) represents a phase shift of 3 to the right.
Vertical shift- the vertical shift is the D value in the image above. When the vertical shift is
positive, you move up. When it is negative you move down. For example, the equation y=sinx-3
is a vertical shift of -3.
Amplitude- Amplitude is represented by the A value in the equation above. to find the
amplitude, you must simply find the distance from the midline to the maximum or minimum
values.
For example, this function
Has an amplitude of 5.
Frequency- The frequency is the b value in the equation above, to find the frequency from a
graph you must find the period (the distance from the beginning of the equation to the end of
one loop) and divide the equation’s pi value by the amplitude. For example, in the graph
The period is 2pi, and since it's a pi equation the pi value is 2pi. So you simply divide 2pi/2pi.
P.S
For equations like y 1=cos(3x+(pi/2)) you must factor out the frequency value. So for x, you
simply divide out 3 to get x. For pi/2 you must divide out 3/1 to get pi/6. This is because when
you divide fractions you find the reciprocal of the number you are dividing by and then multiply
the numerators and denominators. The actual equation then, is: Y=cos 3(x + pi/6)