let 0 be an angle in quadrant IV such that cot0=-(8)/(9). Find the exact value of sin0 and sec0
cot Ø = adjacent/opposite = 8/-9
so let x = 8 and y = -9 in a right-angled triangle in quad IV
r^2 = 64 + 81 = 145
r = √145
sinØ = y/r = -9/√145
secØ = 1/cosØ = r/x = √145/8
To find the exact value of sin0, we can use the identity of cotangent and sine:
cot0 = adjacent / opposite
In quadrant IV, the adjacent side is negative and the opposite side is positive. Therefore, we can represent cot0 as:
cot0 = -adjacent / opposite
Given that cot0 = -8/9, we can equate it to -adjacent / opposite:
-8/9 = -adjacent / opposite
Cross-multiplying, we get:
-8 * opposite = -9 * adjacent
Simplifying, we have:
8 * opposite = 9 * adjacent
Now, let's use the Pythagorean identity to find the hypotenuse:
hypotenuse^2 = opposite^2 + adjacent^2
Since the angle is in quadrant IV, opposite and adjacent will be positive. Substituting the values, we have:
hypotenuse^2 = 8^2 + 9^2
hypotenuse^2 = 64 + 81
hypotenuse^2 = 145
hypotenuse = √145
Now, we can find the value of sine using the equation:
sin0 = opposite / hypotenuse
Substituting the values, we get:
sin0 = (9) / (√145)
sin0 = 9√145 / 145
To find the exact value of sec0, we can use the identity of secant and cosine:
sec0 = 1 / cosine0
To find cosine0, we can use the Pythagorean identity again:
cosine0 = adjacent / hypotenuse
Substituting the values, we get:
cosine0 = (8) / (√145)
Now, we can find the value of sec0 using the equation:
sec0 = 1 / cosine0
Substituting the value of cosine0, we have:
sec0 = 1 / (8 / √145)
sec0 = √145 / 8
To find the exact value of sin(θ) and sec(θ), where cot(θ) = -8/9 and θ is in quadrant IV, we can use the following trigonometric identities:
csc²(θ) = 1 + cot²(θ)
sec²(θ) = 1 + tan²(θ)
First, let's solve for csc²(θ):
csc²(θ) = 1 + cot²(θ) [using the identity]
csc²(θ) = 1 + (-8/9)² [substituting the value of cot(θ)]
csc²(θ) = 1 + 64/81
csc²(θ) = 145/81
Taking the square root of both sides:
csc(θ) = √(145/81)
csc(θ) = √145 / √81
csc(θ) = √145 / 9
Now, to find the exact value of sin(θ), we can take the reciprocal of csc(θ):
sin(θ) = 1 / csc(θ)
sin(θ) = 1 / (√145 / 9) [taking the reciprocal]
sin(θ) = 9 / √145
Finally, let's solve for sec²(θ):
sec²(θ) = 1 + tan²(θ) [using the identity]
sec²(θ) = 1 + (1 / cot²(θ)) [substituting the value of tan(θ)]
sec²(θ) = 1 + (1 / (-8/9)²)
sec²(θ) = 1 + 1 / (64/81)
sec²(θ) = 1 + 81/64
sec²(θ) = (64 + 81) / 64
sec²(θ) = 145/64
Taking the square root of both sides (since sec(θ) > 0):
sec(θ) = √(145/64)
sec(θ) = √145 / √64
sec(θ) = √145 / 8
Therefore, the exact value of sin(θ) is 9/√145 and the exact value of sec(θ) is √145/8.