Need help with 2 Trig problems

Test Preparation ACT Preparation
1

<ol>
<li><p>A right triangle has legs of length 25 sin theta feet and 25 cos theta feet for some angle theta that satisfies 0< theta< 90 degrees. What is the length in feet of the longest side of the triangle?
a. theta
b. 1
c. 5
d. 25
e cannot be determined</p></li>
<li><p>( Im not sure you will understand this without the graph but I will try anyway)
The figure below shows the path of a projectile launched from the ground at an angle of theta. The gorizonal range, R, of this projectile when launched from the ground at a speed of 20 meters per second is modeled by R= 40 sin(2theta).</p></li>
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<p>then there's the diagram of the rocket shooting up and R is moving from the origin to the right...</p>

<p>For this model, the angle measure theta that results in the greatest horizonal range R, is 45 degrees because:
a. 2 sin theta is greater than sin theta
b. sin 90 degrees is as large as sine can get
c. sin 45 degrees is as large as sine can get
d. sin 45 degrees is about 0.707
e. sin 2theta is greater than sin theta</p>

2

<ol>
<li>Its a right triangle so the hypotenuse is the longest length. By definition the hypotenuse=sqrt(a^2+b^2) where a and b are the legs.</li>
</ol>

<p>So, hypotenuse=sqrt(625sin^2(theta)+625cos^2(theta))
h=sqrt(625(sin^2(theta)+cos^2(theta)))
Also, a formula tells us that sin^2(theta)+cos^2(theta)=1.
So h=sqrt(625)=25 (D)</p>

3

<p>oh! thanks! I kinda knew that was the answer but wasn’t really sure why. Any idea on the second one without showing you the graph?</p>

4

<p>I mean for the second one I think the answer is (B) as large as sin 90 can get, but I’m not too sure on that.</p>

<p>The reason it should be is because R=40sin(2theta). The value of sin ranges from -1 to 1. So the max value of sin, giving us the max value of R, is when sin=1. Sin equals 1 when the angle is 90 degrees since sin 90=1. </p>

<p>So sin(2theta)=sin(90) which implies theta=45 will be the max horizontal range.</p>

<p>This all happens because of the fact that the sin value should be 90 since it is the max value of sin. So the answer is B</p>

5

<p>yes, the answer is B.</p>