# Math Help - The terminal arm of angle θ...

1. ## The terminal arm of angle θ...

If the terminal arm of angle θ in standard position passes through the ordered pair (4,5), then...

a) cos θ = √(41)/4

b) sin θ = 5√(41)/41

c) tan θ = 0.555...

d) jakθ = 1/2
My answer key tells me that b) is the answer but I have no clue how to find it. I have been trying for almost an hour now. Could anyone please leave a detailed response, possibly a picture and graph to show how it would look.

2. ## @Math-learner

hi
First fix the points on the co-ordinate plane. you will see that a right triangle is formed when you join the point with
i) the origin
ii) (4,0)
iii) (0,5)

Now, calculate the length of the hypotenuse of this triangle~ a simple pythagoras theorem calculation.

So you will have lengths of all three sides of the triangle~ Next use the ratios of sine, cos and tan. You will see that sin of the angle is given by the ratio of the opposite side : the hypotenuse.

Just substitute the numbers and get the answer. If you aren't able to complete, I'll upload the image bearing the right triangle.

3. Originally Posted by Math-Learner
If the terminal arm of angle θ in standard position passes through the ordered pair (4,5), then...

...Could anyone please leave a detailed response, possibly a picture and graph to show how it would look.
A picture is a great idea, but you can draw it yourself!

Draw your axes, and then draw the dot. Draw a vertical line from the dot to the x-axis. Draw a slanty line from the dot to the origin. This gives you a right triangle.

From the x-value of the dot, you know the length, along the x-axis, of the base of the triangle. Label the base with this value.

From the y-value of the dot, you know the length, parallel to the y-axis, of the height of the triangle. Label the height with this value.

From the lengths of the two legs, you can find the length of the hypotenuse by applying the Pythagorean Theorem. Apply the Theorem, and label the hypotenuse with this value.

Label the angle at the origin as $\theta"$.

Read the values of the sine, cosine, tangent, etc of $\theta$, from your picture. Find the value which matches one of the answer-options. That's your answer!

4. You could also use the trig definitions in terms of x, y, and r. If you have an angle in standard position and the terminal side passes through (x, y), then the trig definitions are
$\sin \theta = \frac{y}{r}$,
$\cos \theta = \frac{x}{r}$,
$\tan \theta = \frac{y}{x}$,
and so on, where
$r = \sqrt{x^2 + y^2}$.

(BTW, What is "jak θ"?)

You have x and y. You can find r, and you can then find the appropriate trig ratios.

Note that in choice b the denominator was rationalized.
$\frac{5}{\sqrt{41}} = \frac{5\sqrt{41}}{41}$.

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