1) Examining those triangles, we can write some trigonometric ratios to find out the measure of theta
1. For this triangle, let's use the tangent and the arctangent of theta. Since we have two legs, the opposite and the adjacent Like this
tan(θ) = 10/13
(θ) = tan^-1(10/13)
θ = 37.568
θ = 37.57º (approximately)
θ = 38º Rounded up to the nearest degree
2) tan(θ) = 3/2
θ = tan ^-1 (3/2)
θ = 56.3099 Round down to the nearest degree
θ = 56º
3) Since in this case, there's a relation between opposite leg and hypotenuse
we'll use sin (θ) and the arcsine sin^-1
sin(θ) = 6/12
θ = sin^-1(6/12)
θ = 30º
4) In this case, we have the adjacent leg and the opposite leg to θ so we'll use tangent tan(θ) and arctan tan^-1 (θ)
tan (θ) = 16/4.7
θ = tan^-1 (16/4.7)
θ = 73.6298 Round up to the nearest degree
θ = 74º
So the answers are
θ = 38º
θ = 56º
θ = 30º
θ = 74º
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The Law of Sines, is valid for any triangle and relates the measure of the triangles' side and the measure of their angles:
Given the angles, A, B, C and with 2 or even 3 given sides, you can find the measure of a missing one.