If tant=−7/24 and π/2 < t < π , find sint , cost , sect , csct , cott .

Answer 1

If tan(t)=−7/24 and π/2 < t < π , find sin(t) , cos(t) , sec(t), csc(t) , cot(t).

Based on SOH-CAH-TOA, tan(x) = opposite/adjascent. So this describes a right triangle with an opposite side of length 7, an adjacent side of length 24, and an unknown hypotenuse. Luckily, due to the Pythagorean Theorem, we can figure that out.

hypotenuse = √ (7² + 24²) = 49.497...

This has a lot of decimal places, so I will keep it in the form √ (7² + 24²), so we don't lose accuracy.

Another thing to keep in mind is that on the unit circle, tangent represents the slope of the hypotenuse (rise/run). So a positive tangent, means that rise and run (or opposite and adjacent) must have the same sign. This means that the angle must be in the first quadrant (both positive) or third quadrant (both negative). We have a negative tangent here, which means the opposite and adjacent sides must have different signs, which means the angle is either in the second or fourth quadrants. The answer specifies that the angle should be between π/2 and π, which confirms that we are working in the 2nd quadrant. This means that the adjacent side (x-coordinate) is negative and the opposite side (y-coordinate) is positive. So this clarifies that the opposite side is 7 and the adjacent side is -24. This is important to specify because it would have been -7 and 24 if it were in the fourth quadrant.

So to summarize, we know the following:

opposite = 7

adjacent = -24

hypotenuse = √ (7² + 24²)

The easiest answer to start with is cot(t), since this is just the reciprocal of tan(t), by definition. So, cot(t) = -24/7.

Using the SOH-CAH-TOA definition of sin(t) = opposite/hypotenuse, sin(t) = 7/√ (7² + 24²). This can be simplified using clever algebra.

sin(t) = 7/√ (7² + 24²)

= 7/√(7²(1 + 24²/7²))

= 7/(7√(1 + 576/49))

= 1/√(1 + 576/49)

= 1/√(49/49 + 576/49)

= 1/√(625/49)

luckily 625 is a perfect square with square root of 25 and 49 is a perfect square with square root of 7, so:

= 1/(25/7)

= 7/25

csc(t) is just the reciprocal of sin(t), by definition, so csc(t) = 25/7

Using the SOH-CAH-TOA definition of cos(t) = adjacent/hypotenuse, cos(t) = -24/√ (7² + 24²). This can also be simplified using clever algebra.

cos(t) = -24 /√ (7² + 24²)

= -24 /√ 24²(7²/24² + 1)

= -24 / 24 √(49/576 + 1)

= -24 / 24 √(49/576 + 576/576)

= -1 / √(625/576)

= -1 / √(25²/24²)

= -24/25

And sec(t) is just the reciprocal of cos(t), so sec(t) = -25/24.

This problem really tests your understanding of the side length definitions of the trigonometric functions as well as the unit circle.

Answer 2

cott = - 24/7; sect = - √(1 + 49/576) = - 25/24; cost = - 24/25; sint = tant·cost = - 7/24·(- 24/25) = 7/25;

csct = 25/7