Trigonometry HELP!!! need it ASAP ?

Question

please help me answer this 2 questions.

1. a picture 7 ft high is placed on a wall with its base 9ft above the level of the eye of the observer. Suppose the observer is x ft from the wall and theta is the radian measure of the angle subtended at the observer's eye by the picture

a) define theta as a function of x

b) find theta when x =10

c) find theta when x =12

d) find theta when x =15

2. a weight is suspended from a spring and vibrating vertically according to the equation y=2sin4pi(T+1/8) where y cm is the directed distance of the weight from its central position T seconds after the start of the motion and positive direction is upward

a) solve the equation for T

b) use the equation to determine the smallest 3 positive values of T for which the weight is 1cm above the central position.

please help me with it...I'm not that good in trigonometry...

Answer 1

Only one problem per question please.

Let

x = distance of observer from wall

α = angle of elevation to from observer's eye to base of picture

θ = angle subtended by observer's eye of picture

α + θ = angle of elevation to from observer's eye to top of picture

Solve for θ.

___________

We have:

tanα = 9/x

tan(α+θ) = (9+7)/x = 16/x

Then by the tangent addition formula we have:

tan(α+θ) = (tanα + tanθ) / [1 - (tanα)(tanθ)]

tan(α+θ) [1 - (tanα)(tanθ)] = tanα + tanθ

tan(α+θ) - tan(α+θ)(tanα)(tanθ) = tanα + tanθ

tan(α+θ) + tanα = tanθ + tan(α+θ)(tanα)(tanθ)

tan(α+θ) + tanα = tanθ[1 + tan(α+θ)(tanα)]

(tan(α+θ) + tanα) / [1 + tan(α+θ)(tanα)] = tanθ

tanθ = (tan(α+θ) + tanα) / [1 + tan(α+θ)(tanα)]

Plug in values.

tanθ = (16/x + 9/x) / [1 + (16/x)(9/x)] = (16x + 9x) / (x² + 16*9)

tanθ = 25x / (x² + 144)

θ = arctan[25x / (x² + 144)]

You can plug in the values for x and solve with a calculator.

Answer 2

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Answer 3

Difficult to solve without making diagrams... and I cant draw here. So sorry...

Answer 4

1) The line from the observers eye to the top of the picture is the hypotenuse of a right triangle. the distance from the observer to the wall is x and the third side of the triangle is 9+7=16 feet.

the tangent of the angle is therefore 16/x.

tan theta = 16/x. a table of trig functions including tangent is included in the links below.