Please help Math Question!?
Given tan θ = -8/5 and sin θ < 0, find sin θ, cos θ, sec θ, csc θ, and cot θ.
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answers:
la console say: tan(θ) = - 8/5 → where: sin(θ) < 0

As tan(θ) is negative and as sin(θ), you can deduce that: cos(θ) > 0

You can see that the angle is in the quadrant IV.



tan(θ) = - 8/5

sin(θ)/cos(θ) = - 8/5

sin(θ) = - (8/5).cos(θ)


Restart from the famous formula:

cos²(θ) + sin²(θ) = 1

cos²(θ) + [- (8/5).cos(θ)]² = 1

cos²(θ) + (64/25).cos²(θ) = 1

25.cos²(θ) + 64.cos²(θ) = 25

89.cos²(θ) = 25

cos²(θ) = 25/89

cos²(θ) = (± 5/√89)²

cos(θ) = ± 5/√89 → recall: cos(θ) > 0

cos(θ) = 5/√89

sec(θ) = 1/cos(θ)

sec(θ) = (√89)/5



Recall:

sin(θ) = - (8/5).cos(θ) → we've just seen that: cos(θ) = 5/√89

sin(θ) = - (8/5).(5/√89)

sin(θ) = - (8 * 5)/(5 * √89)

sin(θ) = - 8/√89

csc(θ) = 1/sin(θ)

csc(θ) = - (√89)/8



cot(θ) = 1/tan(θ)

cot(θ) = 1/(- 8/5)

cot(θ) = - 5/8
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Como say: Presentation is unclear so will take a guess at :-
tan ∅ = -- 8/5
4 th quadrant
sin ∅ = -- 8/√89
cos ∅ = -- 5/√89
tan ∅ = -- 8/5
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electron1 say: Tan θ = -1.6

If you use a calculator to determine the angle, you will see that the angle is approximately -58˚. This means the angle is approximately 58˚ counter clockwise from the positive x axis. This shows the angle is in the fourth quadrant.

http://math2.org/math/algebra/functions/...

If you go to the website above, you will see how the six trigonometry functions are related to each other. Sin θ and and csc θ are inverse functions. Cos θ and sec θ are inverse functions.


Tan θ = y/x

This is the ratio of the height and base of a right triangle. To determine the hypotenuse of the right triangle, use the following equation.

Hypotenuse = √(Height^2 + Base^2) = √(-8^2 + 5^2) = √89

Sin θ = -8 ÷ √89
Cos θ = 5 ÷ √89
Sec θ = √89 ÷ -5
Csc θ = √89 ÷ -8
Cot θ = 5/-8 = -0.625

I hope this is helpful for you.
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Johnathan say: The reference triangle is 5/8/sqrt(89), as by the Pythagorean Theorem, 5^2 + 8^2 = (sqrt(89))^2.

You have tan Θ negative and sin Θ also negative. So cos Θ is positive (> 0), meaning Θ is in quadrant 4.

sin Θ = -8 / sqrt(89), or -8 sqrt(89) / 89.

cos Θ = 5 / sqrt(89), or 5 sqrt(89) / 89.

cot Θ = -5/8 (tan and cot are reciprocals, so this function should be the easiest to figure).

sec Θ = sqrt(89) / 5 (just invert cos Θ).

csc Θ = -sqrt(89) / 8 (just invert sin Θ).
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ted s say: conditions mean that the angle is in the 4th quadrant so the reference triangle is { 5 , - 8 , √89 }...use it
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