Answer:
1) secФ = √2 ⇒ 2nd answer
2) The best identity is 1 + cot²Ф = csc²Ф ⇒ 4th answer
3) The expression 1/(-3/8) is equivalent to cotФ ⇒ 1st answer
Step-by-step explanation:
* The angle Ф lies in the 4th quadrant because 2π/3 < Ф < 2π
∴ The values of cosФ and secФ only positive
* Use the identity sec²Ф = tan²Ф + 1
∵ tanФ = -1
∴ sec²Ф = (-1)² + 1 = 1 + 1 = 2 ⇒ take √ in both sides
∴ secФ = ± √2
∵ Ф lies in the 4th quadrant
∴ secФ = √2
* Remember if we have the value of cotФ, we can use the identity
cot²Ф + 1 = csc²Ф to find the value of cscФ
∵ cscФ = 1/sinФ
∴ sinФ = 1/cosФ
* In the problem Fatma know that cotФ = 4/7
∵ She wants to know the value of sinФ
∴ She can use the identity cot²Ф + 1 = csc²Ф
* The best identity is cot²Ф + 1 = csc²Ф
* We know that cotФ = 1/tanФ
∵ tanФ = -3/8
∴ cotФ = 1/(-3/8)
∴ The expression 1/(-3/8) is equivalent to cotФ
