Question

find all triginometric values of θ with the given conditions:
cos θ= -15/17 , sin θ > 0

Answer 1

Remember that cos is negative in the second, and third quadrants (the two left ones), and sin is poisitive in the first, and second quadrants (the top two).

Answer 2

draw a right triangle. one side is 15 and the hypotenuse is 17 so the other side must be
\[\sqrt{17^2-15^2}=8\]

Answer 3

now just take ratios being mindful that you are in quadrant 2 because cosine is negative and sine is positive

Answer 4

\[\sin(\theta)=\frac{8}{17}\]
\[\tan(\theta)=-\frac{8}{15}\]
\[\sec(\theta)=-\frac{17}{15}\]
\[\csc(\theta)=\frac{17}{8}\]
\[\cot(\theta)=-\frac{15}{8}\]

Answer 5

With \(cos(x)=\frac{15}{17}\). If you perform the inverse cos operation, you'll get \(x = 0.49 rad \: / \: 28.07^{o}\) as your reference angle. Since we're only looking at the second and third quadrants, since cos is negative, you'll get two answers, which are \(\pi \pm 0.49 \: / \: 180 \pm 28.07^{o}\).
For \(sin\theta > 0\), you have to take all quadrants where \(sin\theta\) is postitive, i.e. quadrants one, and two. These cover the angles \(0 \le \theta \le \pi \: / \: 0 \le \theta \le 180^{o}\)