Question

The terminal arm of an angle is standard position passes through 𝑃(−5,5).

a. Sketch a diagram of an angle in standard position.

b. Determine the length of the hypotenuse.

c. Determine the primary trigonometric ratios to three decimal places.

Answer 1

(a) Diagram of the angle in Standard position

(b) Length of the hypotenuse (Hyp)

$Hyp = \sqrt{(-5)^2 + (5)^2}\\ \qquad = \sqrt{25 + 25}\\ \qquad = \sqrt{50}\\ Hyp = 5\sqrt{2}$

(c) The primary trigonometric ratios are sin, cos and tan.

Since the terminal arm lies in 2nd quadrant, we have to take positive signs for sin, while others are negative.

opp = 5, adj = -5

(i) sin

$\sin{\alpha} = \dfrac{opp}{hyp}=\dfrac{5}{5\sqrt{2}}=\dfrac{1}{\sqrt{2}}=\dfrac{1}{1.4142}\\ \bold{\sin{\alpha} = 0.707}$

(ii) cos

$\cos{\alpha} = \dfrac{adj}{hyp}=\dfrac{-5}{5\sqrt{2}}=-\dfrac{1}{\sqrt{2}}=-\dfrac{1}{1.4142}\\ \bold{\cos{\alpha} = -0.707}$

(iii) tan

$\tan{\alpha} = \dfrac{opp}{adj}=\dfrac{5}{-5}=\bold{-1}$