<ol>
<li>If sin A = cos B, then which of the following must be true?
(A)A = B
(B)A = 2B
(C)A = B + 45
(D)A = 90 − B
(E)A = B + 180</li>
</ol>
<p>The answer is (D), but could someone explain this mathematically, and how to do it mathematically, without plugging in? Plugging in isn't my way of working, I'm not used to it. :/</p>
<ol>
<li>Runner A travels a feet every minute. Runner B travels b feet every second. In one hour, runner A travels how much farther than runner B, in feet?</li>
</ol>
<p>(A)a − 60b
(B)a2 − 60b2
(C)360a − b
(D)60(a − b)
(E)60(a − 60b)</p>
<p>Question 1: there’s a formula : cos(90-x) = sin x (you can use cos(a-b) formula with a=90 to prove it)
so we can write this equation : sin A = cos B <=> sin A = sin (90-B). We conclude that A = 90-B
Question 2: runner B runs b feets per second so he runs 60b feets per minute. if he runs for an hour=60minutes , the distance will be Speed x Time, the speed is in feets per minute so the time should be in minutes. Thus, he runs 60 (time) * 60 b (speed) = 3600 b. runner A runs a feets per minute so he runs 60 (time) * a (speed) = 60a. The difference between them is Distance run by A - Distance run by B = 60a-3600b <=> 60(a-60b)
Question 3: the expression ( x^2 + 3x - 4 ) / ( 2x^2 + 10x + 8 ) is undefined when its denominator equals 0. So we have to find the values of x that make 2x^2 + 10x + 8 = 0. It’s a quadratic equation so you can solve it by calculating the discriminant or you can use your calculator (I have a scientific calculator, I can’t help with graphing calculators)
it will give you two values : -1 and -4</p>
<p>For #3, x = -1 and x = -4 result in the denominator being equal to 0. However x = -4 results in the fraction 0/0, which is usually said to be indeterminate, rather than undefined.</p>
<ol>
<li><p>Recall the definition of sine and cosine:
Let us have a right triangle ABC, angle C = 90 degrees. Then sin A = BC/AB = cos B and cos A = AC/AB = sin B.
As we know, in a right triangle the sum of acute angles equals 90 degrees.
A+B=90.
From this equality we get A = 90 – B. The correct answer is D)</p></li>
<li><p>Runner A travels a feet in a minute and in one hour he travels 60a feet (1 hour = 60 minutes). Runner B travels b feet in a second it is the 60 feet in a minute (1 minute = 60 seconds), and it is 360 feet in one hour (1 hour = 60 minutes). To find the difference we should subtract the distance that runner B pass from the pass that runner A pass.
60a – 360b=60(a-60b).
The correct answer is E)</p></li>
<li><p>The fraction is undefined at the points where denominator equals zero. So we should just find the solutions of equation 2x^2+10x+8=0. Dividing this equation by 2 we get x^2+5x+4=0.
This is a quadratic equation. So
D=5^2-4<em>1</em>4=25-16=9
x<em>1=(-5-sqrt(9))/2=(-5-3)/2=-4
and
x</em>2=(-5+sqrt(9))/2=(-5+3)/2=1
At these points the denominator of our fraction equals 0, therefore, for these values of x the expression is undefined.
The correct answer is D)</p></li>
</ol>
<p>For 1, there’s a trig rule that for any trig function f(x) and the cofunction of the trig function, g(x), f(x)=g(90-x). Basically,
sin(x)=cos(90-x); cos (x)=sin(90-x)
tan(x)=cot(90-x);cot(x)=tan(90-x)
sec(x)=csc(90-x);sec(x)=csc(90-x)</p>
<p>sine and COsine are cofunctions, so sine of one angle will equal the cosine of the complement of the angle. </p>
<p>2: Runner b travels b feet per second, so every minute, runner B travels 60*B feet. Runner A travels A feet per minute. One hour has 60 minutes, so runner B travels 60(60B) feet, and Runner A travels 60A feet. The difference is simply 60A-3600B, but it can be simplified into 60(A-60B).</p>
<ol>
<li>Functions are undefined when the denominator is equal to zero. So, simply find the x values for which the denominator equals zero by factoring the denominator. You could also plug in answer choices, but that takes longer.</li>
</ol>
<p>Number 1 is a trig rule that you can prove using 30, 60 , 90 tringles, number 2, multiply 60 by 60 twice (runner B) then subtract it from 60 (runner A) then factorise.</p>
very clear explanations