Trigonometry Intro Flashcards by Zqpu Man

State the properties of similar triangles

Corresponding angles are the same
Corresponding sides are proportional
The ratio of their areas are proportional to the square of the ratio of their corresponding sides

How well did you know this?

Not at all

Perfectly

What is SOH CAH TOA?

sinx = o/h
cosx = a/h
tanx = o/a

cscx = h/o
secx = h/a
cotx = a/o

How well did you know this?

Not at all

Perfectly

What is the angle of elevation and depression?

Visualize it

How well did you know this?

Not at all

Perfectly

Describe an angle in standard position

The vertex is at origin (0, 0). The terminal arm is a rotation x degrees of the initial arm (positive x-axis), about the origin.

If x > 0, rotation counter clock wise
If x < 0, rotation clock wise

How well did you know this?

Not at all

Perfectly

What’s the formula for a circle centered at the origin?

r^2 = x^2 + y^2
You can rearrange this formula to find r, the radius

How well did you know this?

Not at all

Perfectly

What are co-terminal angles?

Co-terminal angles share a terminal arm and differ by a multiple of 360 degrees. They have the same ratios

How well did you know this?

Not at all

Perfectly

What’s the formula to finding co-terminal angles?

x +- k360 degrees, where k is a natural number

How well did you know this?

Not at all

Perfectly

State the special angles and thei ratios

sin30 = 1/2
cos30 = √3/2
tan30 = 1/√3
sin60 = √3/2
cos60 = 1/2
tan60 = √3
sin45 = 1/√2
cos45 = 1/√2
tan45 = 1
sin0 = 0
cos0 = 1
tan0 = 1

How well did you know this?

Not at all

Perfectly

What are reference angles and why are they useful?

An acute angle formed between the terminal arm and x-axis. It’s used to relate angles in any quadrant back to Q1.

How well did you know this?

Not at all

Perfectly

How can you find the reference angle in each quadrant?

Q1: x
Q2: 180 - x
Q3: x - 180
Q4: 360 - x

How well did you know this?

Not at all

Perfectly

How can you find quadrant angles given the reference angle x?

Q1: x
Q2: 180 - x
Q3: 180 + x
Q4: 360 - x

How well did you know this?

Not at all

Perfectly

What is the CAST rule?

It shows where each ratio is positive

How well did you know this?

Not at all

Perfectly

How many solutions does each ratio have between 0 and 360 degrees (one circle)

2 primary solutions

How well did you know this?

Not at all

Perfectly

What are related angles?

Related angles have the same side ratios

How well did you know this?

Not at all

Perfectly

What are the Q1 related angles?

sinx = -sin(-x)
cosx = cos(-x)
tanx = -tan(-x)

How well did you know this?

Not at all

Perfectly

What are the Q2 related angles?

sin(180-x) = sinx
cos(180-x) = -cosx
tan(180-x) = -tanx

How well did you know this?

Not at all

Perfectly

What are the Q3 related angles?

sin(180+x) = -sinx
cos(180+x) = -cosx
tan(180+x) = tanx

How well did you know this?

Not at all

Perfectly

When do you use sine and cosine law?

When the triangle has no right angle

How well did you know this?

Not at all

Perfectly

What are the Q4 related angles?

sin(360-x) = -sinx
cos(360-x) = cosx
tan(360-x) = -tanx

How well did you know this?

Not at all

Perfectly

State sine law and when you use it

sinA/a = sinB/b = sinC/c
a/sinA = b/sinB = c/sinC

You use it when you have an AAS or SSA triangle

How well did you know this?

Not at all

Perfectly

State cosine law and when you use it

cosA = (b^2 + c^2 - a^2) / 2bc
a^2 = b^2 + c^2 - 2bccosA

You use it when you have an SAS or SSS triangle

How well did you know this?

Not at all

Perfectly

What’s the formula for the area of a triangle in terms of sin?

b(asinC) / 2

asinC is the height

How well did you know this?

Not at all

Perfectly

Explain the ambiguous case

When you have an SSA triangle, it creates ambiguity.
If a > b, then you have 1 triangle

If a < b, then you have multiple cases:
a = bsinA, 1 right angle triangle
a < bsinA, no triangle
a > bsinA, 2 triangles, ambigous

How well did you know this?

Not at all

Perfectly

How do you write a true bearing?

Start from N and measure clockwise. 3 digits needed
Example: 030 degrees

How well did you know this?

Not at all

Perfectly

How do you write a compass bearing?

Start from N or S and measure acute angle, state E or W Example: N30E

What are radians?

An angle subtended at centre by an arc equal in length to the radius has a measure of 1 radian. It's a unit for measuring angles.

What's the formula for a central angle in a circle in radians?

x = a/r Where a is the arc length and r is the radius

How much radians is 360 degrees?

2π

How much radians is 180 degrees?

π (3.14)

How do you convert degrees to rad?

Multiply by π / 180 degrees

How do you convert rad to degrees?

Multiply by 180 degree / π

What's the formula for the area of a sector?

(1/2)(r^2)(x)

What's the formula for the perimeter of a sector?

2r + a

What's the area of the area of a segment?

(1/2)(r^2)(x - sinx)

What is angular velocity?

The angular velocity of a rotating object is the rate at which the central angle changes w/ respect to change

How much is 1 revolution in rad and deg?

2π and 360 degrees

How do you calculate the central angle in an angular velocity question?

x = (angular velocity)(time)

How do you convert minutes to seconds?

Multiply by 60 seconds / 1 minute

What does the arc length represent in an angular velocity question?

The actual distance traveled along the curved path (circumference) by a point on a rotating object

What's a periodic function?

It contains a repeated pattern

What's a cycle?

It's one iteration of a pattern

What's the period?

The length of the cycle

How do you find the period of a sin/cos/tan graph?

For sin/cos: P = 2π / |b| For tan: P = π / |b|

What is amplitude and how do you find it?

The distance between the max and the min (only for sinusoidal functions), |a| Ampl = (max - min) / 2

What's a sinusoidal function?

It's sinx and cosx

What's a phase shift?

Horizotnal translations

What's the sinusoidal axis and how do you find it?

The axis around which a sinusoidal function oscillates S.A = (max + min) / 2

What's the general formula of a transformed sinusoidal function?

f(x) = asinb(x - d) + c |a| is the amplitude, d is the phase shift, c is used in the S.A y = c.

What's the pattern of a sine graph?

mid, max, mid, min, mid

What are the characteristics of a base sine function?

D: all real numbers R: [-1, 1] Ampl: 1 Period: 2π S.A: y = 0 Phase shift: none

What's the pattern of a cosine graph?

max, mid, min, mid, max

What are the characteristics of a base cosine function?

D: all real numbers R: [-1, 1] Ampl: 1 Period: 2π S.A: y = 0 Phase shift: none

How are the cos and sin functions related?

cosx = sin(x + π/2) The cosine function is a phase shift of the sine function

What are the characteristics of a tangent function?

D: x is not equal to π/2 + kπ R: all real numbers Ampl: none Period: π S.A: none Phase shift: none

What are some graphing tips?

Split the period into 4 parts by multiplying it by 1/4. Try to create a scale that makes phase shifts easy to sketch

What is arcsinx, arccosx, arctanx?

The inverse of sinx, cosx, and tanx

How do you graph arcsinx and what are its characteristics?

1. Restrict the domain to allow the inverse to exist 2. Swap x and y points For domain restricted sinx: D: [-π/2, π/2] R: [-1, 1] For arcsinx: D: [-1, 1] R: [-π/2, π/2]

How do you graph arccosx and what are its characteristics?

1. Restrict the domain 2. Swap x and y points For domain restricted cosx: D: [0, π] R: [-1, 1] For arccosx: D: [-1, 1] R: [0, π]

How do you graph arctanx and what are its characteristics?

1. Restrict the domain 2. Swap x and y points 3. Determine end behaviour For domain restricted tanx: D: (-π/2, π/2) R: all real numbers V.A: x = -π/2, π/2 For arctanx: D: all real numbers R: (-π/2, π/2) H.A: y = -π/2, π/2

When does sinx = 0?

At x = nπ

When does cosx = 0?

At x = π /2 + nπ

When does tanx = 0?

At x = π /2 + nπ

When does sinx = 1?

At x = π /2 + 2nπ

When does cosx = 1?

At x = 2nπ

When does tanx = 1?

At x = π /4 + nπ

When does sinx = -1?

At x = 3π/2 + 2nπ

When does cosx = -1?

At x = π + 2nπ

When does tanx = -1?

At x = 3π/4 + nπ

How do you find all solutions in a given range (like -π to 3π/2)?

Find the solutions in one period, then add or subtract the period repeatedly until you get all solutions that fall within the given range

When is csc x undefined? When is sec x undefined? When is cot x undefined?

csc x is undefined when sin x = 0 sec x is undefined when cos x = 0 cot x is undefined when tan x = 0 (or sin x = 0)

When does csc x = 1? When does sec x = 1?

csc x = 1 when sin x = 1 sec x = 1 when cos x = 1

When does cotx = 1?

cot x = 1 when cos x = sin x, which happens at x = π/4 + nπW

What is sin(-x) equal to?

sin(-x) = -sin(x) Sine is an odd function

What is cos(-x) equal to?

cos(-x) = cos(x) Cosine is an even function

What is tan(-x) equal to?

tan(-x) = -tan(x) Tangent is an odd function

Trigonometry Intro Flashcards

(75 cards)