Trig Test #1 5.1-5.4

acute angle

0° < θ < 90°

trig functions of quadrantal angles

0°, 90°, 180°, 270° 0º= (1,0) 90º= (0,1) 180º= (-1,0) 270º= (0,-1)

evaluating trig functions

1. find the reference angle 2. find the value of the trig function 3. give the correct sign to the function value

straight angle

180°

special right triangles

45-45-90 30-60-90

right angle

90°

obtuse angle

90° < θ < 180°

trig functions of any angle

P= (x,y) which really means (cosθ,sinθ) x=cosθ y=sinθ sinθ= y/r cosθ= x/y tanθ=y/x cscθ= r/y secθ= r/x cotθ= x/y

What degrees and do All systems belongs to?

Quadrant I * 0° < θ < 90° (meaning this quadrant is between 0° and 90°) *all quadrants (sin, cos, tan, csc, sec, and cot are all positive) Quadrant II * 90° < θ 180° (meaning this quadrant is between 90° and 180°) * ONLY sinθ and cscθ are positive Quadrant III * 180° < θ < 270° (meaning this quadrant is between 180° and 270°) * ONLY tanθ and cotθ are positive Quadrant IV * 270°<θ<360° (meaning this quadrant is between 270° and 360°) * ONLY cosθ and secθ are postive

quadrantal angle

an angle is called this if it is a multiple of 90° (0, 90, 180, 270, 360,...)

pythagorean theorem

a²+b²=c² where a and b are the legs and c is the hypotenuse This equation holds for any right triangle

even and odd trig functions

cosine and secant are EVEN functions cos (-t)= cos t sec (-t)= sec t sine, cosecant, tangent, and cotangent are ODD functions sin (-t)= -sin t csc (-t)= -csc t tan (-t)= -tan t cot (-t)= -cot t

right triangle trigonometry

hypotenuse: the side that is opposite to the right angle opposite: the leg that the angle opens into adjacent: the leg that the angle is attached to

standard position

if one side is at 0° and the other is at its angle value going counterclockwise (if it is a positive angle) or clockwise (if it is a negative angle).

the unit circle

is a circle with a radius of r=1 that is defined by the parametric equations sin t = y cos t= x tan t= y/x where t is the angle measure

what is the formula you need to use if you are converting degrees to radians?

multiply the degrees by π/180° (pi divided by 180°)

what is the formula you need to use if you are converting radians to degrees?

multiply the radians by 180°/π (180° divided by pi)

what is the formula for the radius (distance from the origin)?

r= √x² + y² (so the radius equals the square root of x squared plus y squared)

what is the formula for length of a circular arc?

s= rθ *θ must be in radians r=radians θ= degrees (but remember to change to radians)

periodic properties of the sine and cosine functions

sin(t+ 2π)= sin t cos(t+2π)= cos t

pythagorean identities

sin²θ + cos²θ= 1 1+ tan²θ= sec²θ 1+cot²θ= csc²θ

reciprocal identities

sinθ= 1/cscθ cosθ= 1/secθ tanθ= 1/cotθ cscθ= 1/sinθ secθ= 1/cosθ cotθ= 1/tanθ

cofunction identities

sinθ= cos(90°-θ) cosθ= sin(90°-θ) tanθ= cot(90°-θ) cotθ= tan(90°-θ) secθ= csc(90°-θ) cscθ= sec(90°-θ) if θ is in radians, replace 90° with π/2

The 6 Trig Functions sin, cos, tan, csc, sec, and cot

sinθ= opposite/hypotenuse cosθ= adjacent/hypotenuse tanθ= opposite/adjacent cscθ= hypotenuse/opposite secθ= hypotenuse/adjacent cotθ= adjacent/opposite

periodic properties of tangent and cotangent functions

tan(t+π)= tan t cot(t+π)= cot t

quotient identities

tanθ= sinθ/cosθ cotθ= cosθ/sinθ

coterminal angles

two angles are coterminal if their difference is a multiple of 360° (or 2π if they are in radians). for example: a) 530° 530°-360°= 170° b) 13π/6 13π/6 - 2π= 13π/6-12π/6= π/6

what is the formula for central angle?

θ= s/r s= length of arc r= radius

reference angle

a reference angle is the angle formed by the original angle and the x-axis. they are always positive.