unit 4: trig triangles

trig n triangles

soh-cah-toa

soh: opposite/ hypotenuse

cah: adjacent/ hypotenuse

toa: opposite/ adjacent

altittude

a line between a triangle that creates two triangles angles. the way you solve is you set up a proportion. _ /x = x/_

hypotenuse

the longest side of any angle; always opposite to the 90∘ angle.

opposite

the side of a triangle opposite to the angle or “θ.”

adjacent

the side right next to “θ.”

sine

sin(θ) = opposite / hypotenuse

cos

cos(θ) = adjacent / hypotenuse

tan

tan(θ) = opposite / adjacent

similar triangles

we will see two different triangles, one is bigger and one is smaller: they are “similar.”

even though they have different magnitudes, they have the same ratios: so the sides are scaled up or down by the same amount, and they will have the same angles (∠)

smaller leg

In right-triangle trigonometry, a leg is always shorter than the hypotenuse, meaning that identifying or calculating a leg from a hypotenuse will always result in a smaller number.

Explain in words what the statement tan(11.31) = 0.2 means.

in a right triangle, with a θ of 11.31, the adjacent size is this much larger than the opposite side because of the 0.2 ratio.

triangle sum theorum

the sum of angle measures must equal out to 180∘

A + B + C = 180∘

angles in right triangles are determined by…

the ratio of the sides of the triangle.

inverse trig functions

these input a ratio and output a ratio.

find an angle where sinx = cosx

45 degrees

radians

a sector with an arc length that has an arc

angles and degrees

the space between two radian points

relation between radians and degrees

2π radian = 360°

so

π radian = 180°

parts of triangles

theta: the opposite will always be opposite to this

area of right triangle: ½ x base x height

Arc Length Formula

S = r0

S = arc length

r = radius

0 = angle of the radius

terminal side of angle

the line that rotates from the fixed initial side.

convert radians to degrees

degrees = radian x 180/π

convert degrees to radians

radians = degree x π/180

transitive property

if a first element relates to a second, and the second relates to a third, then the first relates to the third

special triangle: 45 45 90:

Ratio: Leg : Leg : Hypotenuse = 1 : 1 : √2

Sides: If the leg is x , the other leg is x , and the hypotenuse is x √2.

Finding Legs from Hypotenuse: Divide the hypotenuse by √2.

Special triangle: 30 60 90

Ratio: Short Leg : Long Leg : Hypotenuse = 1 : √3 : 2

Sides: If the short leg (opposite 30∘) is x.

Hypotenuse (opposite 90 degr) = 2x

long leg (opposite 60 degr) = x√3

cosine and sine relationship

cos is x: as you increase, x decreases

sin is y: as you increase, y increases

Pythagorean theorem

formula for right triangles: a² + b² = c², where 'a' and 'b' are the two shorter sides (legs) and 'c' is the longest side (hypotenuse) opposite the right angle.

ordered pairs of unit circle

X and Y: negativity

reference angle

positive 90 degree angle formed between the terminal side of a given angle and the x-axis.

• it is always 90 degrees or less

Coterminal angles

angles in standard position that share the same terminal side, differing by a full 360 degrees.

trigonometric standard form

y = a cos (

y = a sin (

amplitude

period

midline