If sin theta =a/b and theta lies in first quadrant, find all t-ratios.
given:
sinØ = a/b
so y = a , x = ?
r = b
r^2 = x^2 + y^2
b^2 = x^2 + a^2
since we are in quad I,
x = +√(b^2 - a^2)
now use your definitions to state the trig ratios
To find all the trigonometric ratios (t-ratios) of an angle, we can use the given value of the sine ratio.
Given: sin(theta) = a/b, where theta lies in the first quadrant.
In the first quadrant, all trigonometric ratios are positive. Here are the different trigonometric ratios:
1. Sine (sin): Already given as sin(theta) = a/b.
2. Cosine (cos): In the first quadrant, cos(theta) is the square root of 1 minus the square of sine. Therefore, cos(theta) = sqrt(1 - (a/b)^2).
3. Tangent (tan): tan(theta) is calculated by dividing sine by cosine. So, tan(theta) = sin(theta) / cos(theta), which is equal to (a/b) / sqrt(1 - (a/b)^2).
Thus, the t-ratios for the given condition are:
sin(theta) = a/b,
cos(theta) = sqrt(1 - (a/b)^2),
tan(theta) = (a/b) / sqrt(1 - (a/b)^2).
These ratios hold true for any angle theta in the first quadrant, given the condition sin(theta) = a/b.