On the other hand sin theta is negative because the y value is negative in the 4th quadrant. So in the 4 quadrant since x is positive cosine is positive since cos theta = x. Tan theta = y/x and then you could figure out where each trigonometric function is positive based on y and x values in the unit circle. T-Tangent is the only positive trigonometric function in the third quadrantĬ- Cosine is the only positive trigonometric function in the 4 quadrant.Īlternatively know that cos theta = x, sin theta = y, S- Sine is the only positive trigonometric function in the second quadrant Whether a trigonometric function is positive is indicated by ASTCĪ- All are positive in the first quadrant To find sin(36°) we can use the double angle formula Which can be simplified to (25 − 10√5)∕5².Īgain, 18° lies in the first quadrant, so tan(18°) is positive, and we get To get a somewhat neat expression for tan(18°) we begin by finding Multiplying both sides by 𝑥 and then subtracting 1 from both sides, we getīecause it is isosceles, we know that its height bisects both the 36° angle and the base, thus splitting the triangle into two congruent right triangles –Įither of which having one angle being 18° with the opposite side having length 1∕2Īnd a hypotenuse of length 𝑥 = (1 + √5)∕2.īecause 18° lies in the first quadrant, we know that cos(18°) is positive:
Since this triangle is similar to the original triangle, we can set up the equation We can figure out that this triangle has a base of length 𝑥 − 1, while the legs are of length 1.
This splits the triangle into two new isosceles triangles, one of which is a 72-72-36 triangle. Then we draw the bisector to one of the 72° angles. We let the base of this isosceles triangle have length 1, and the legs have length 𝑥. The two angles you mention, 𝜋∕5 = 36° and 𝜋∕10 = 18°, we can actually find exact expressions for that aren't too complicated.
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