Answer:
1) tanФ = 0.8392 ⇒ 2nd answer
2) cosФ = 20/29 ⇒ 3rd answer
3) cosФ = -√2/2 , cotФ = -1 ⇒ first answer
Step-by-step explanation:
* At first check the quadrant of the angle
∵ 180° < Ф < 270°
∴ Ф lies on the third quadrant
* Remember in the 3rd quadrant tanФ and cotФ only positive
∵ sinФ ≅ -0.7660
∵ tan²Ф = sec²Ф - 1
∵ secФ = 1/sinФ
∵ secФ = 1/-0.7660
∴ sec²Ф = (1/-0.7660)²
* Substitute this value in the equation
∴ tan²Ф = (1/-0.7660)² - 1 = 0.70428594 ⇒ take √ for both sides
∴ tanФ ≅ ± 0.8392
∵ Ф lies on the 3rd quadrant
∴ tanФ = 0.8392
* At first check the quadrant of the angle
∵ 0° < Ф < 90°
∴ Ф lies on the first quadrant
* Remember in the 1st quadrant all are positive
∵ sinФ = 21/29
∵ sin²Ф + cos²Ф = 1
∴ (21/29)² + cos²Ф = 1 ⇒ subtract (21/29)² from both sides
∴ cos²Ф = 1 - (21/29)² = 400/841 ⇒ take √ in both sides
∴ cosФ = ± 20/29
* Because Ф lies in the 1st quadrant
∴ cosФ = 20/29
* Remember that when the point is at the terminal side of angle Ф
∴ Its x-coordinate is cosФ
∴ Its y-coordinate is sinФ
* The point is (-√2/2 , √2/2)
∵ x-coordinate is negative ad y-coordinate is positive
∴ the point is on the 2nd quadrant
∴ 90° < Ф < 180°
∴ The values of sinФ and cscФ only positive
* From previous we know that:
∴ cosФ = -√2/2
∴ sinФ = √2/2
∵ cotФ = cosФ/sinФ
∴ cotФ = -√2/2 ÷ √2/2 = -√2/2 × 2/√2 = -√2/√2 = -1
* cosФ = -√2/2 , cotФ = -1
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