Trig - Chapter 7.5 & 7.6

y = arctan(x)

-π/2 < y < π/2 I + IV

y = arcsin(x)

-π/2 ≤ y ≤ π/2 I + IV

y = arccsc(x)

-π/2 ≤ y ≤ π/2 y ≠ 0 I + IV

0° = 30° = 45° = 60° = 90° =

0 π/6 π/4 π/3 π/2

y = arccot(x)

0 < y < π I + II

y = arccos(x)

0 ≤ y ≤ π I + II

y = arcsec(x)

0 ≤ y ≤ π y ≠ π/2 I + II

Approximate values

1. Be in the correct mode on your calculator 2. Plug values into calculator 3. If it is cscØ, secØ or cotØ, plug the reciprocal angle in with sinØ, cosØ or tanØ

Finding the exact values in degrees and radians

1. Write down the range 2. Find the quadrants 3. Circle the correct quadrant that matches the positive or negative value 4. If it is cscØ, secØ, or tanØ, flip the fraction and rationalize 5. Graph your reference triangle 6. Plug in you're trig values in radians or degrees 7. Check that your answer is in the correct quadrant and range

Find missing trig functions

1. You're given one trig function + range 2. Find the quadrant it's in 3. Use x2 + y2 = r2 to find x, y, and r (Use ± when finding square rt!!! r is always positive!!!) 4. Plug into reference triangle 5. Plug variables into missing trig functions and simplify Look at 7.5 notes

If 0 ≤ y ≤ π and you're in the second quadrant

180° - the reference angle to get your final answer

Inverse sin can also look like

Arcsin - Goes for other trig functions

If you get an error

Check if you plugged your function correctly into your calculator

TanØ = y/x

CotØ = x/y Calculator: 1 / tanØ

SinØ = y/r

CscØ = r/y Calculator: 1 / sinØ

Find approximate values

Depending on if the angle is in radians or degrees, you must be in the correct mode on your calculator in order for the answer to be correct

SinØ and cosØ have a range of -1 ≤ y ≤ 1.

Expect an ERROR if you plug into your calculator sin-1 or cos-1 of a number greater than 1 or less than -1

If -π/2 ≤ y ≤ π/2 is your range and your in the IV quadrant

Find the reference angle and make it negative, and that will be your final answer

SinØ + cscØ

I = + II = + III = - IV = -

TanØ + cotØ

I = + II = - III = + IV = -

CosØ + secØ

I = + II = - III = - IV = +

Exact values

Just like before, use reference angle and plug values in for variables on reference triangle to get the correct answer

Exact values =

No decimal!!!

Exact values: Angle is in quadrant II

Reference angle = 180° - Ø

Exact values: Angle is in quadrant IV

Reference angle = 360° - Ø

Exact values: Angle is in quadrant I

Reference angle = Ø

Exact values: Angle is in quadrant III

Reference angle = Ø - 180°

CosØ = x/r

SecØ = r/x Calculator: 1 / cosØ

Trig function(angle) = Trig function-1(value =

Value Angle

If the value is 0

You're on the axis

If the value is 1 for sinØ, cosØ and reciprocals

You're on the axis

If you take inverse tan or inverse cot of 1 or -1

Your reference angle is 45°

Exact values

x2 + y2 = r2

Find the approximate and exact value of cos(tan-1(-2/3))

• The mode doesn't matter for these problems A.V. = 0.83 • 0.83 is a value, not an angle!!! E.V. -π/2 < y < π/2 I + IV ---> IV because -2/3 is negative x2 + y2 = r2 (3)2 + (-2)2 = r2 9 + 4 = r2 √13 = √r2 ±√13 = r +√13 = r Plug into reference triangle CosØ = 3/√13 x √13/√13 CosØ = 3√13/13 E.V. = 3√13/13