Pg. 323/361 Homework• Pg. 362 #39 – 47 odd, 51, 52

Memorize Trig. Info

• #2 QIV #4 QIII #6 QIII• #8 -250° #10 470°• #12 338°, 698°, -382°, -742° #14 210°, 570°, -510°, -870°• #16 11π/4, -5π/2 #18 29π/6, -19π/6• #20 π/4, 9π/4, -15π/4, -23π/4 #22 67°, 157°• #24 78°, 168° #26 5π/12, 11π/12 #28 π/14, 4π/7• #30 90°, π/2 #32 135°, 3π/4 #34 0°, 0, 2π• #36 -120°, -2π/3 #38 -600°, -10π/3 #40 -1260°, -7π• #42 π/3 #44 7π/6 #46 11π/6• #48 59π/90 #50 249π/180 #52 810/π• #1 #2 #3 8/7• #4 7/8 #5 #6• #7• #8 11/15 #9 #10• #11 #12 #13 15/11

8113

7113

1137

1138

2 2615

152 26

2 2611

112 26

113 1137 8 7 8, , , , ,8 7 8 7113 113

6.5 Trig Functions of an Acute Angle

Trig Functions• The six trig functions of any

angle 0° < Ɵ < 90° are defined as follows:

• Also, we know that from the basic three trig functions:

sin cot

cos sec

tan csc

opp adj

hyp opp

adj hyp

hyp adj

opp hyp

adj opp

1cot

tan1

seccos1

cscsin

6.5 Trig Functions of an Acute Angle

Triangle Trig• We use right triangles

because they allow us to use the Pythagorean Theorem, which makes solving a much easier process!

• Because 30°, 45° and 60° occur frequently, we will learn and memorize those triangles!!

• If c = 2, determine the lengths of a and b and find the six trig values at 30° and 60°.

6.5 Trig Functions of an Acute Angle

More Trig Triangles• If a = 1, determine the

length of c and find the six trig values.

Cofunctions of Complementary Angles

• If Ɵ is any acute trig angle, a trig function value of Ɵ is equal to the cofunction of the complement of Ɵ, as follows:

sin cos cot tan2 2

cos sin sec csc2 2

tan cot csc sec2 2

6.5 Trig Functions of an Acute Angle

Example:• Show that

are cofunctions. • Let Ɵ be an acute angle

such that sin Ɵ = 5/6. Find all the trig functions of Ɵ.

• One angle of a right triangle measures 38°, and the hypotenuse has length of 17. Find the measures of the remaining sides.

sin cos2

6.5 Trig Functions of an Acute Angle

Example:• They hypotenuse and one

leg of a triangle measure 12 and 7 respectively. Find the measure of angle Ɵ formed by these two sides.

Trig Functions of Any Angle• Let Ɵ be an angle in

standard position, P(x, y) a point other than the origin on the terminal side of P, and r = The six trig values of Ɵ are defined as follows:sin cos tan

sec csc cot

y x y

r r xr r x

x y y

2 2x y

6.5 Trig Functions of an Acute Angle

Example:• Find the values of the six

trig functions at an angle Ɵ in standard form with point P(4, 7) on its terminal side.

• **Note** If an angle does not appear acute in the first quadrant, you can work in any of the four quadrants to make the angle acute!!

• Find the values of all six trig functions and the angle of measure of angle Ɵ, where P(-3, 2) is a point on the terminal side of Ɵ.

• Find all the values of all six trig functions for the quadrantal angles of:

30, , , , 22 2