The point P(-4,9) is on the terminal arm of ∠A
(a) Find the primary trigonometric ratios (sinA, cosA, tanA) in exact values.
(b) Determine the measure of ∠A to the nearest degree.
you have
x = -4
y = 9
r = √(x^2+y^2) = √97
Now recall that
cosθ = x/r = -4/√97
sinθ = y/r
tanθ = y/x
and so on.
(b) so use the absolute value of any of those trig functions to find the reference angle Ø. In QII you want θ=π-Ø
To find the values of the primary trigonometric ratios (sinA, cosA, tanA) in exact values, we first need to determine the lengths of the sides of the right triangle formed by the point P(-4,9) on the coordinate plane.
Using the distance formula, we can calculate the lengths of the sides as follows:
a = √((-4)^2 + 9^2) = √(16 + 81) = √97
Now, we can find the values of the trigonometric ratios:
(a) The primary trigonometric ratios are defined as follows:
sinA = opposite/hypotenuse = y-coordinate/radius
cosA = adjacent/hypotenuse = x-coordinate/radius
tanA = opposite/adjacent = y-coordinate/x-coordinate
sinA = 9/√97
cosA = -4/√97 (negative because the point P is in the third quadrant)
tanA = 9/-4 = -9/4
(b) To determine the measure of ∠A to the nearest degree, we can use the inverse tangent function (tan^-1) to find the angle whose tangent is -9/4.
∠A = tan^-1(-9/4) ≈ -67.38 degrees (rounded to the nearest degree)
Therefore, the measure of ∠A to the nearest degree is approximately -67 degrees.
To find the primary trigonometric ratios (sinA, cosA, tanA) for ∠A, we need to determine the lengths of the sides in the triangle formed by the given point P(-4,9) on the terminal arm of ∠A.
Let's start with finding the hypotenuse (r) of the triangle. The hypotenuse is the distance between the origin (0,0) and the point P(-4,9), which can be calculated using the distance formula:
r = sqrt((-4 - 0)^2 + (9 - 0)^2)
= sqrt((-4)^2 + (9)^2)
= sqrt(16 + 81)
= sqrt(97)
Now, we can find the values of sinA, cosA, and tanA:
(a) sinA = opposite / hypotenuse = 9 / sqrt(97)
cosA = adjacent / hypotenuse = -4 / sqrt(97) (since the x-coordinate is negative)
tanA = sinA / cosA = (9 / sqrt(97)) / (-4 / sqrt(97)) = -9 / 4
(b) To determine the measure of ∠A to the nearest degree, we can use the inverse trigonometric functions. In this case, we can use the inverse tangent (arctan) function to find the angle value:
∠A = arctan(tanA)
= arctan(-9/4)
To calculate this, you can use a calculator or online tool that provides the value of arctan(-9/4). The result is approximately -66.04 degrees (to two decimal places).
Therefore, the measure of ∠A to the nearest degree is -66 degrees.