In triangle PQR, angle Q= 90 degrees, angle P= 60 degrees and angle R= 30 degrees. PR= 1 unit. Extend side QR to T such that PR= RT. Join PT.
Calculate the exact measure of angle T.
Determine the exact value of lengths needed to find tan T and then find tan T.
∆PRT is isosceles
∠PRT is 180-30 = 150
So, ∠P and ∠T are (180-150)/2 = 15°
PT^2 = 1^2 + 1^2 - 2(1)(1)cos150°
= 2+√3
Consider ∆PQT and you can get tan T
To find the measure of angle T in triangle PQR, we need to consider the fact that PT is a straight line.
Since angle Q is a right angle, the sum of angles P and R must be 90 degrees. So, angle R is 90 - 60 = 30 degrees.
Now, let's analyze the triangle again. We have angle P = 60 degrees and angle R = 30 degrees, which means that angle PTR is 180 - (60 + 30) = 90 degrees.
Therefore, angle T is also 90 degrees, as it forms a straight line with angle PTR.
To find the lengths needed to calculate tan T, we can consider triangle PTR.
We know that PR = 1 unit and PT = PR (as PR = RT). Let's call the length of PT x.
Using the Pythagorean theorem in triangle PTR, we have:
PT^2 + PR^2 = RT^2
x^2 + 1^2 = (2x)^2
x^2 + 1 = 4x^2
3x^2 = 1
x^2 = 1/3
x = √(1/3)
Now, we can calculate the value of tan T using triangle PTR.
tan T = PT/RT
tan T = (√(1/3))/(2√(1/3))
tan T = (1/√3)/(2/√3)
tan T = 1/2
Therefore, the exact value of tan T is 1/2.
To calculate the exact measure of angle T, we can use the fact that the sum of angles in a triangle is 180 degrees. Since we know angle Q is 90 degrees and angle P is 60 degrees, we can subtract these angles from 180 degrees to find angle T.
Measure of angle T = 180 - (90 + 60)
Measure of angle T = 180 - 150
Measure of angle T = 30 degrees
Now let's find the exact values of the lengths needed to calculate tan T. We can use the trigonometric ratios in the triangle to find the lengths.
Since angle R is 30 degrees, we have a 30-60-90 triangle. In a 30-60-90 triangle, the ratios of the side lengths are as follows:
- The side opposite the 30 degrees angle is always half the length of the hypotenuse.
- The side opposite the 60 degrees angle is always (√3/2) times the length of the hypotenuse.
- The hypotenuse is always twice the length of the side opposite the 30 degrees angle.
From the given information, we know that PR = RT = 1 unit. Therefore, PT is the hypotenuse of the triangle.
Let's find the length of PT:
PT = 2 * PR
PT = 2
Now, we can find the length of QR (which is also the length of RT):
QR = (√3/2) * PT
QR = (√3/2) * 2
QR = √3
Finally, we can calculate the value of tan T:
tan T = (RT / QR)
tan T = (1 / √3)
tan T = (√3 / 3)
Therefore, the exact measure of angle T is 30 degrees, and the exact value of tan T is (√3 / 3).