Answer:
1) x is negative and y is positive ⇒ last answer
2) cotФ = -12/35 ⇒ second answer
3) The right identity is cot²Ф - csc²Ф = -1 ⇒ last answer
Step-by-step explanation:
* For any point (x , y) lies on the terminal side of the angle Ф
in standard position
* x = cosФ and y = sinФ
- If Ф in the first quadrant, then x , y are positive
∴ All trigonometry functions are positive
- If Ф in the second quadrant, then x is negative , y is positive
∴ sinФ only is positive
- If Ф in the third quadrant, then x is negative , y is negative
∴ tanФ only is positive
- If Ф in the fourth quadrant, then x is positive , y is negative
∴ cosФ only is positive
* Lets solve the problems
∵ Ф = 3π/4 ⇒ (135°)
∴ It lies on the second quadrant
∴ x is negative and y is positive
* Lets revise the reciprocal of sinФ, cosФ and tanФ
- cscФ = 1/sinФ
- secФ = 1/cosФ
- cotФ = 1/tanФ
∵ secФ = -37/12
∴ cosФ = -12/37
∵ π/2 < Ф < π
∴ Ф lies on the second quadrant
∴ cotФ is negative values
∵ tan²Ф = sec²Ф - 1
∵ secФ = -37/12
∴ tan²Ф = (-37/12)² - 1 = 1225/144 ⇒ take√ for both sides
∴ tanФ = ± 35/12
∵ cotФ = ± 12/35
∵ cotФ is negative value
∴ cotФ = -12/35
* In the standard position of the angle Ф the terminal
of it lies on the unit circle O
- By using Pythagorean theorem
∵ x² + y² = 1
∵ x = cosФ and y = sinФ
∴ cos²Ф + sin²Ф = 1 ⇒ (1)
∴ cos²Ф = 1 - sin²Ф
∴ sin²Ф = 1 - cos²Ф
* Divide (1) by cos²Ф
∴ cos²Ф/cos²Ф + sin²Ф/cos²Ф = 1/cos²Ф
* Remember sin²Ф/cos²Ф = tan²Ф and 1/cos²Ф = sec²Ф
∴ 1 + tan²Ф = sec²Ф ⇒ (2) ⇒ subtract 1 from both sides
∴ tan²Ф = sec²Ф - 1 ⇒ subtract sec²Ф from both sides
∴ tan²Ф - sec²Ф = -1
* Divide (1) by sin²Ф
∴ cos²Ф/sin²Ф + sin²Ф/si²Ф = 1/sin²Ф
* Remember cos²Ф/sin²Ф = cot²Ф and 1/sin²Ф = csc²Ф
∴ cot²Ф + 1 = csc²Ф ⇒ (3) ⇒ subtract 1 from both sides
∴ cot²Ф = csc²Ф - 1 ⇒ subtract csc²Ф from both sides
∴ cot²Ф - csc²Ф = -1
* The right identity is cot²Ф - csc²Ф = -1
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