Question

Identify the triangle that contains an acute angle for which the sine and cosine ratios are equal. 1. Triangle A B C has angle measures 50 degrees, 40 degrees, and 90 degrees. 2. Triangle A B C has angle measures 45 degrees, 45 degrees, 90 degrees. 3. The lengths of sides A C and C B are congruent. 4. Triangle A B C has angle measures 68 degrees, 22 degrees, and 90 degrees. 5. Triangle A B C has angle measures 60 degrees, 30 degrees, and 90 degrees.

Answer 1

Answer:

The correct option: (2) Triangle ABC that has angle measures 45°, 45° and 90°.

Step-by-step explanation:

It is provided that a triangle ABC has an acute angle for which the sine and cosine ratios are equal to 1.

Let the acute angle be m∠A.

For the sine and cosine ratio of m∠A to be equal to 1, the value of Sine of m∠A should be same as value of Cosine of m∠A.

The above predicament is possible for only one acute angle, i.e. 45°, since the value of Sin 45° and Cos 45° is,  

                                 $Sin\ 45^{o} =Cos\ 45^{o} = \frac{1}{\sqrt{2} }$

So for acute angle 45° the ratio of Sin 45° and Cos 45° is:

                                         $\frac{Sin\ 45^{o}}{Cos\ 45^{o}} = \frac{\frac{1}{\sqrt{2} } }{\frac{1}{\sqrt{2} } } = 1$

Hence one of the angles of a triangle is, m∠A = 45°.

Comparing with the options provided the triangle is,

Triangle ABC that has angle measures 45°, 45° and 90°.

Thus, the provided triangle is a right angled isosceles triangle, since it has two similar angles.