Algebra: Lap 8, Trig functions

How to remember the unit circle values for the first quadrant:

- The left hand trick: pretend you left hand represents the first quadrant of the unit circle, with each finger representing an angle (pinky = 90, thumb = 0). Fold down the finger that is the angle you want. The coordinates for that angle will be (√uppers /2, √lowers /2) - From 30º to 60º the x coordinate goes from √3, √2, √1 (over 2) while the y coordinate goes from √1, √2, √3 (over 2) -All unit circle values the don't lie on the axis have a denominator of 2 for both the x coordinate and the y coordinate

Why is the angle direction significant?

- if the angle direction is counterclockwise, then the angle is positive -if the angle direction is clockwise, then the angle is negative -angles have multiple revolutions

For what values can we restrict the tan function to make it one-to-one?

-90º< angle <90º

For what values can we restrict the sin function to make it one-to-one?

-90º≤ angle ≤90º

reciprocal

-a number that, when multiplied by another number, gives 1 -switches the numerator and denominator -The reciprocal of sine is cosecant -the reciprocal of cosine is secant -the reciprocal of tan is cotangent

What are the unit circle angles for the rest of the unit circle?

-all other angles are a reflection of the quadrant 1 angles -quadrantal angles = 0º, 90º, 180º, 270º, 360º -30º angle versions = 30º, 150º, 210º, 330º -45º angle versions = 45º, 135º, 225º, 315º -60º angle versions = 60º, 120º, 240º, 300º -All angles in the unit circle correspond to one of the 30, 45, or 60 angles of quadrant one based on how far away the new angle is from the X AXIS

What is an angle?

-an angle is formed by rotating a ray around its endpoint -all angles have an initial ray, a terminal ray, and an angle direction

What is the unit circle?

A circle centered on the origin with a radius of 1 and circumference of 2π

Trigonometric function

A function whose rule is given by a trigonometric ratio (angle measure is the input, the rule is the sin / cos/ tan / etc. pattern and the output is the result which is the ratio that follows the pattern)

what is a one-to-one function?

A one to one function must pass the vertical and horizontal line test. No x or y coordinate can repeat. If and only if a function is one to one, then its inverse will also be a function. However, if a function fails the horizontal line test (two x values have the same y value) then its inverse will not be a function

How did we find these coordinate values for each angle?

-based on the rules for special triangles -for a 30º angle, the radius (hypotenuse) is 1 (2x), so the y value (opposite side) is 1/2 (x) and the x value (adjacent side) is √3/2 (x√3) -for a 45º angle, the radius (hypotenuse) is 1 (x√2), so the y value (opposite side) is √2/2 (x) and the x value (adjacent side) is √2/2 (x) -for a 60º angle, the radius (hypotenuse) is 1 (2x), so the y value (opposite side) is √3/2 (x√3) and the x value (adjacent side) is 1/2 (x) -90º (0,1) and 0º (1,0) are on the axis

Standard position of an angle

-initial side is on the positive x axis -the vertex is at the origin

What are the two types of coterminal angles?

-the coterminal angle's direction is in the same direction as the original angle, just with more revolutions -the coterminal angle's direction is the opposite direction as the original angle

revolution

-the number of times the terminal ray swings around the center -revolutions can be in either direction and can be positive or negative -one full revolution has 360º

What are the degrees of the first quadrant of the unit circle?

0º, 30º, 45º, 60º, 90º

For what values can we restrict the cos function to make it one-to-one?

0º≤ angle ≤180º

tan (45º)

1

What are the coordinates of the first quadrant of the unit circle?

1. 0º = (1,0) 2. 30º = (√3/2, 1/2) 3. 45º = (√2/2, √2/2) 4. 60º = (1/2, √3/2) 5. 90º = (0,1)

How to find the area of a triangle:

1. Area= (0.5)bc sin(A) 2. Area = (0.5)(base)(height) 3. Area = √(s)(s-a)(s-b)(s-c)

How to solve trig equations when looking for a missing angle:

1. Identify which trig function is in question 2. Set up the formula for finding an angle using the inverse trig function 3. Solve the formula 4. Restrict the answer to a given range in the problem

How does the unit circle help us find trig ratio values?

1. The sine of an angle equals the y coordinate that the angle passes through divided by the radius 2. The cosine of an angle equals the x coordinate that the angle passes through divided by the radius 3. The tan of an angle equals the y coordinate that an angle passes through divided by the x coordinate that an angle passes though

What are the three possible cases when solving triangles using the law of sines or cosines?

1. no triangle 2. one triangle 3. two triangles

How to use law of sines:

1. set up the law of sines using the known side and its known angle, and then the other known side and its unknown angle 2. Solve for the unknown angle 3. If you end up with an invalid trig function, there is no triangle 4. If the unknown angle is smaller than the known angle, then there is one triangle. If the unknown angle is bigger than the known angle, then there are two triangles 5. Once you find the unknown angle, find the third unknown angle 6. Set up law of sines to find the final unknown side 7. If there were two triangles, subtract THE AMBIGUOUS ANGLE from 180 to find the other option. Use this other option for steps 5-6 to find the other option for the other unknown angle and side

How to convert from parametric equations to x&y equations

1. solve x equation for t 2. Sub it in for t in the y equation 3. Solve for y

What are the two parts of the unit circle?

1. the angles 2. the coordinates of the endpoints of the angles

cos (60º)

1/2

sin (30º)

1/2

What are the two special right triangles?

45-45-90 and 30-60-90

parametric equations

A set of two equations in terms of time that, together, give you the ordered pair of the location of a moving object at a certain time

When do we use law of sines?

ASS, SAA, ASA (any type of triangle - acute, obtuse, or right)

What are the coordinates of angles in the unit circle beyond quadrant 1?

All other coordinates can be found by simply switching the sign of the quadrant 1 coordinates of the angle that this angle in symmetrical to (ex: 30º is (√3/2, 1/2) so 150º (which is the "30" angle of the second quadrant) will be (-√3/2, 1/2))

Trig ratio theorem for ordered pairs on the unit circle

If P is a point where the terminal side of an angle tº in standard position meets the unit circle then P has the coordinates (cos(t), sin(t)) (note: this only works on the unit circle because the radius is 1 so cos = x and sin = y. On other circles, cos = x/r and sin = y/r)

What is trigonometry?

The study of triangle measurements and angles

Why do we need to divide by r when finding sin and cos based on the coordinates of the unit circle?

The value of the trig function of an angle remains the same regardless of the size of the circle. In order to preserve the same value for different x and y values that do vary based on size, we must divide by r because it grows and shrink proportionately to the growth and shrink of the x or y value when the size of a circle in increased or decreased (however, for a unit circle only, the radius is 1 so it doesn't matter)

Why is the domain and range of the inverse trig functions significant?

This prevents the inverse trig functions from spitting out multiple answers because we must restrict the final answer to be within the limits

As of now, we can input an angle and use our trig ratios to find a ratio. But what if we don't know the angle?

We can use inverse trig functions

solving a right triangle

finding all angle measures and all side lengths of a right triangle through use of either trig ratios or the Pythagorean theorem

solving a triangle

finding the measures of all the unknown sides and angles of a triangle

When is there one triangle?

if you end up with a trig function where the ratio is NOT between -1 and 1 (i.e. cosA=6) (this is because sin and cos trig functions have a range that is always between -1 and 1)

trig ratio theorem for a given point on ANY circle

let t be a real number and (x,y) be any point except the origin on the terminal angle tº in standard position. Then: sin(t) = y/r cos(t) = x/r tan (t) = y/x where r = √(x²+y²) is the distance from (x,y) to the origin

Sine

sin (ϴ) = opposite / hypotenuse

Pythagorean identity

sin²θ+cos²θ=1 (the square of the sin of an angle plus the square of the cosine of that angle will ALWAYS equal 1)

What are the inverses for each trig value?

sin⁻¹ = arcsin cos⁻¹ = arccos tan⁻¹ = arctan

angle of elevation

the angle between the ground and the line of sight when an observer looks upward

angle of depression

the angle between the line of sight and the horizontal when an observer looks downward

quadrantal angle

the five angles whose terminal side lies on an axis when in standard position 0º = no revolution 90º= 1/4 revolution 180º= 1/2 revolution 270º= 3/4 revolution 360º= 1 full revolution note: quadrantal angles also include any coterminals

opposite leg

the leg across from a given acute angle in a right triangle

adjacent leg

the leg that touches a given acute angle in a right triangle

Hypotnuse

the side opposite of a right angle in a right triangle (the longest side)

law of cosines

the square of any side equals the sum of the squares of the other two sides minus the product of twice the other two sides and the cosine of the angle a²=b²+c²-2bcCosA b²=a²+c²-2acCosB c²=a²+b²-2abCosC

why is the unit circle important?

the unit circle (or "trig" circle) is significant because it allows us to apply the ratios of sin, cos, and tan to ANY angle between 0º and 360º. Now, we are not limited to angles less than 90º nor are we limited to right triangles

coterminal angles

two angles in standard position that have the same initial side and the same terminal side, but different measures

What is the formula for finding the coterminal angle of a given angle?

xº + 360ºn -x is the given angle measure -n is the number of revolutions you want the new coterminal angle to have (can be + or -) -360 is the number of degrees in a circle and thus stands for one revolution

Solving: Sinθ = #

θ = sin⁻¹(#) + 360ºn or θ = 180º- sin⁻¹(#) + 360ºn

Solving: tanθ=#

θ=tan⁻¹(#)+180ºn

solving: cosθ=#

θ=±cos⁻¹(#) +360ºn

What "ϴ" mean?

ϴ (called "theta") is a Greek character that is used to signify an unknown angle measurement

cos (45º)

√2/2

sin (45º)

√2/2

tan (60º)

√3

cos (30º)

√3/2

sin (60º)

√3/2

tan (30º)

√3/3

law of sines

For any ∆ABC with side lengths a,b, and c, sinA÷a = sinB÷b = sinC÷c

A regular trig function (sin cos tan) or a reciprocal trig function (csc, sec, cot) all INPUT AN ANGLE and OUTPUT A RATIO

An inverse trig function (sin⁻¹ cos⁻¹ tan⁻¹) INPUTS A RATIO and OUTPUTS AN ANGLE

What is the domain and range of each inverse trig function:

Arcsin: D= -1≤ ratio ≤1, R= -90º≤ angle ≤90º Arccos: D= -1≤ ratio ≤1, R= 0º≤ angle ≤180º Arctan: D=-∞<ratio<∞, R= -90º< angle <90º

cotangent

COT (ϴ) = adjacent / opposite

Cosecant

CSC (ϴ) = hypotenuse / opposite

Cosine

Cos (ϴ) = adjacent / hypotenuse

How to find trig ratios for all the other quadrants?

Find the trig ratio for the corresponding first quadrant angle and adjust the sign (All Students Take Calculus: first quadrant = all are positive, 2nd quadrant = sine and its reciprocal are positive, 3rd quadrant = tan and its reciprocal are positive, 4th quadrant = cosine and its reciprocal are positive) Basically, all trig ratios for angles that are symmetrical about the main axes are the same value except for its sign (+/-)

What are the two ways to solve a triangle with NO right angle?

Law of sines and law of cosines

Are the functions for sin, cos, and tan one-to-one?

No.... but we need their inverses!! So in order to FORCE the trig functions to be one-to-one, we restrict their domains

When is there two triangles?

ONLY ASS TRIANGLES CAN HAVE TWO TRIANGLES!!!

Arc

Part of a circle connecting two points on the circle.

Secant

SEC (ϴ) = hypotenuse / adjacent

When do we use law of Cosines?

SSS, SAS (only oblique triangles, any triangle that is not right triangles)

SOH CAH TOA

Sine, Cosine, Tangent = SOH CAH TOA Cosecant, Secant, Cotangent = reciprocals of SOH CAH TOA

Tangent

Tan (ϴ) = opposite / adjacent

sector

The part of a circle that looks like a piece of pie. A sector is bounded by 2 radii and an arc of the circle.

Right triangle

a triangle with one right angle, signified by 90º or a box

What is an inverse function?

an inverse function results when the domain and range of a function have been switched