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