Unit 7: Basic Trigonometry

cosine (cos)

Adjacent / Hypotenuse (CAH)

cotangent (cot)

Adjacent / Opposite

Notice:

All six trig functions can be positive or negative depending on the quadrant (beta) is located in. (You can remember this by ASTC --> *All* (Q1) *s*tudents (Q2) *t*ake (Q3) *c*rack (Q4)... each letter stands for which trig functions are positive in each quadrant)

Quadrant 1

All trig functions are positive

45-45-90 triangle

An isosceles right triangle (or you can substitute the x's for ones)

cosecant (csc)

Hypotenuse / Opposite

tangent (tan)

Opposite / Adjacent (TOA)

Reference Triangles

The triangle formed by dropping a perpendicular line from the terminal ray of a standard position to the x-axis. (remember: your triangle should be part of a bowtie)

SOH-CAH-TOA

Trigonometric relationships (sine, cosine, tangent)

Co-terminal angles

angles that share the same initial and terminal sides (two angles that have the same initial side and the same terminal side, but have different measures). *Add or subtract 360º (as need) to find a co-terminal angle* Examples: 30º & -330º // 270º & -90º // (π/3) & (-5π/3)

a NEGATIVE rotation is

clockwise (when the angle is negative it is a negative rotation)

sine and cosine are referred to as

cofunctions

cot (beta) =

cos (beta) / sin (beta)

Reciprocal Trig Functions

cosecant, secant, cotangent

secant is the reciprocal of

cosine

A POSITIVE rotation is

counterclockwise

when a question asks you to solve a triangle, its asking you to

find all of its sides AND angles

Inverse trig functions are used to

find an *angle.* (Recall: Inverse functions SWITCH the inputs and the outputs... if sin (beta) = x, the sin^-1 (x) = (beta)... the same goes for cosine and tangent) [sin^-1(beta) = arcsin (beta) ... cos^-1(beta) = arccos (beta) ... tan^-1(beta) = arctan (beta)]

In a coordinate plane, an angle can be formed by fixing one ray, called the

initial side, and rotating the other ray, called the terminal side, about the vertex.

Angles can be measured using a different unit of measure than degrees. One radian is the measure of an angle (beta) in standard position whose rays

intercept an arc equal to the length of the radius of the circle

1 degree can be divided into small units of measure called

minutes and seconds. - 1 degree = 60 minutes... - 1 minute = 60 seconds... (ex: 18 degrees, 52 minutes, 3 seconds is written as 18º 52' 3") [When calculating minutes, after you do, for example, sin^-1(beta), keep all of (beta)'s decimals. Then, take away/subtract the whole number - do not round the whole number when you list the degree it represents - and multiply the decimal by 60 - which will give you minutes. Do the same if you're calculating seconds]

Quadrant 2

only sin & csc are positive (cos, sec, tan, & cot are negative)

Quadrant 3

only tan & cot are positive (sin, csc, cos, & sec are all negative)

An angle is in standard position when its vertex is the

origin and the initial side lies on the positive x-axis

cosecant is the reciprocal of

sine

Angles that measure 30, 45, and 60 occur frequently in trig. We can use

special right triangles to find the value of the six functions for these angles

cotangent is the reciprocal of

tangent

If you know 2 sides of a right triangle, you can use

the Pythagorean theorem to find the measure of the third side

Radians are based on the

unit circle, which is a circle with a radius of 1 whose center is the origin. The equation of they circle is .... x^2 + y ^2 = 1

If you divide sin/cos

you get tan

To convert between degrees and radians use this equation:

( radians / π ) = ( degrees / 180 ) and cross multiply. To convert from degree measure to radians, multiply by *( π / 180 )* To convert from radian measure to degrees, multiply by *( 180 / π )*

(x,y) =

(cos (beta), sin (beta)) ... *This is only true when the radius is 1* (This is only true if the reference triangle is on the edge of the x-axis)

To find the exact value of a trig function

1.) find the reference angle... 2.) determine the exact value of the trig function for the reference angle... 3.) determine the sign (positive or negative) based on the *quadrant* of the original angle... Helpful: (π/3) = 60º ... (π/6) = 30º... (π/4) = 45º... 45, 45, 90 triangle & 30, 60, 90 triangle...

The coordinate plane has

4 quadrants

secant (sec)

Hypotenuse / Adjacent

sine (sin)

Opposite / Hypotenuse (SOH)

Reference Angle

an acute angle formed by the terminal side of (beta) and the closes *x-axis* (the reference angle is always positive) (If the angle given is a quadrantal angle, use the unit circle to find out what the reference angle is equal to) For any nonquadrantal angle (beta), 0 < (beta) < 360º (0 < (beta) < 2π), its reference angle is defined as follows: - if the angle is in Quadrant 1, the reference angle is the measure of (beta)... - if the angle is in Quadrant 2, the reference angle is the measure of 180º - (beta)... - if the angle is in Quadrant 3, the reference angle is the measure of (beta) - 180º... - if the angle is in Quadrant 4, the reference angle is the measure of 360º - (beta)

Remember, anytime the value of sin(beta) or cos(beta) is a positive real number between 0 and 1, your graphing calculator will display (beta) as

an acute angle in the first quadrant... What if we want (beta) in another quadrant? Then we need to use reference angles. (then use the rules of reference angle's to add or subtract degrees in order to find the angle that is in the quadrant you want) (ex: sin(beta) = 0.9652, Quadrant II --> (beta) = sin^-1(0.9652) --> (beta) = 75º (Ref. angle) --> 180 - 75 = 105º)

Pathagorean Theorem

a² + b² = c² (where c is always the hypotenuse)

When two angles add up to 90º they are called

complementary (in a diagram of a 3,4,5 right triangle, the two unknown angle measures add up to 90.... angle a = 90 - b.... angle b = 90 - a)

Ratios of a right triangles sides are used to

define trigonometric functions

Quadrant 4

only cos & sec are positive (sin, csc, tan, & cot are all negative)

If the terminal side of an angle coincides with a coordinate axis (lies on the lines of the axis), the angle is referred to as a

quadrantal angle

sector equation

s = 𝜃r (s = length of the sector // r = radius // 𝜃 = central angle) *(always measured in radians)*

tan (beta) =

sin (beta) / cos (beta)

cofunctions

sine and cosine are cofunctions // tangent and cotangent are cofunctions // secant and cosecant are cofunctions. (subtract from 90 to find each function - angle - value in terms of its cofunction) (If the question asks to solve for x, add both sides and set the whole thing equal to 90)

You must rationalize the values of trig functions of a triangle if

there is a radical in the denominator of the fraction by multiplying the numerator and the denominator by the radical number

cos (beta) =

x (coordinate) ... -1 ≤ cos (beta) ≤ 1 ... The x-coordinate of the ordered pair corresponds with the cosine of the angle. (*This is only true when the radius is 1*)

sin (beta) =

y (coordinate) ... -1 ≤ sin (beta) ≤ 1 ... The y-coordinate of the ordered pair corresponds with the sine of the angle. (*This is only true when the radius is 1*)

When given a trigonometric function value, use

your calculator to find the angle measure by using sin^-1, cos^-1, or tan^-1 key. (make sure calculator is in degree mode)

Unit Circle

a circle with a radius of 1, whose center is at the origin and has an equation of x^2 + y^2 = 1

sector

a region of a circle that is bounded by two radii and the arc of the circle. The central angle (beta) of a sector is the angle formed by two radii. (ignore formula in picture)

30-60-90 triangle

a right triangle that consists of a right angle, a 30 degree angle, and a 60 degree angle (or you can substitute the x's for ones)