Find the exact value of the trigonometric expression when

1 answer:

5 0

Part of the value of sin(u) is cut off; I suspect it should be either sin(u) = -5/13 or sin(u) = -12/13, since (5, 12, 13) is a Pythagorean triple. I'll assume -5/13.

Expand the tan expression using the angle sum identities for sin and cos :

tan(u + v) = sin(u + v) / cos(u + v)

tan(u + v) = [sin(u) cos(v) + cos(u) sin(v)] / [cos(u) cos(v) - sin(u) sin(v)]

Since both u and v are in Quadrant III, we know that each of sin(u), cos(u), sin(v), and cos(v) are negative.

Recall that for all x,

cos²(x) + sin²(x) = 1

and it follows that

cos(u) = - √(1 - sin²(u)) = -12/13

sin(v) = - √(1 - cos²(v)) = -3/5

Then putting everything together, we have

tan(u + v)

= [(-5/13) • (-4/5) + (-12/13) • (-3/5)] / [(-12/13) • (-4/5) - (-5/13) • (-3/5)]

= 56/33

(or, if sin(u) = -12/13, then tan(u + v) = -63/16)

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Write the point-slope equation of the line that passes through(3,5) and (7,1)

Liono4ka [1.6K]

Answer: y - 1 = –(x –7)

or y –5 = –1(x –3)

The attachment shows the graph of these equations (identical)

Step-by-step explanation:

The basic form for a point-slope equation is

y - b = m(x - a)

a is the x-value from the given coordinate, b is the y-value form the given coordinate, m is the slope. x and y are the variables in the function.

To find the slope, m, use the given coordinates. Get the difference in the y-values and divide by the difference in the x-values. This is "rise over run."

Subtract the first coordinate from the second coordinate.

Given coordinate points: (3,5) and (7,1) substitute values:

m= 1 –5 / 7 –3 becomes m= –4/4 so the slope, m = –4/4

That is –1 The "1" is "understood," so only the negative sign represents the slope in the equation

To write the equation: Take the basic form, substitute b and a values from either given coordinate. And use the value of m we just calculated.

y - b = m(x - a) Using the second coordinate: (7,1)

y - 1 = –(x –7)

or using the first coordinate:

y –5 = –(x –3)