The point P(k, 24) is 25 units from the origin. If P lies on the terminal arm of an angle, θ, in standard position, 0° ≤ θ < 360°, determine
a) the measure(s) of θ
b) the sine, cosine, and tangent ratios for θ
+118723 I immediately recognise 25,24,7 as being a pythagorean triad. (Not that you need to use this)
Draw the triangle, in the number plane (the radius it the hypotenuse, the centre of he circle is the origin.)
the y value is positive so theta is in the 1st or 2nd quadrant.
$$\theta = arcsine(24/25)\qquad or \qquad \theta = 180-arcsine(24/25)$$
arc sin is the same as inverse sin.
$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{24}}}{{\mathtt{25}}}}\right)} = {\mathtt{73.739\: \!795\: \!291\: \!688^{\circ}}}$$
OR
$${\mathtt{180}}{\mathtt{\,-\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{24}}}{{\mathtt{25}}}}\right)} = {\mathtt{106.260\: \!204\: \!708\: \!312}}$$
$$sin \theta = 24/25
cos \theta = \pm 7/25
tan \theta = \pm 24/7$$
+118723 I immediately recognise 25,24,7 as being a pythagorean triad. (Not that you need to use this)
Draw the triangle, in the number plane (the radius it the hypotenuse, the centre of he circle is the origin.)
the y value is positive so theta is in the 1st or 2nd quadrant.
$$\theta = arcsine(24/25)\qquad or \qquad \theta = 180-arcsine(24/25)$$
arc sin is the same as inverse sin.
$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{24}}}{{\mathtt{25}}}}\right)} = {\mathtt{73.739\: \!795\: \!291\: \!688^{\circ}}}$$
OR
$${\mathtt{180}}{\mathtt{\,-\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{24}}}{{\mathtt{25}}}}\right)} = {\mathtt{106.260\: \!204\: \!708\: \!312}}$$
$$sin \theta = 24/25
cos \theta = \pm 7/25
tan \theta = \pm 24/7$$
