Answer:
Option D. Yes, because angles measuring [tex](\theta)[/tex] and [tex](90\°-\theta)[/tex] are complements
Step-by-step explanation:
we know that
Two angles are complementary if their sum is equal to [tex]90[/tex] degrees
and
If two angles are complementary, then the function sine of one angle is equal to the function cosine of the other angle
therefore
In this problem
[tex](\theta)+(90\°-\theta)=90\°[/tex]
so
[tex](\theta)[/tex] and [tex](90\°-\theta)[/tex] -----> are complementary angles
hence
[tex]sin(\theta)=cos(90\°-\theta)[/tex]
The correct option is [tex]\boxed{{\mathbf{option D}}}[/tex] .
Further explanation:
Given:
It is given that [tex]\sin\theta=\cos\left({90-\theta }\right)[/tex] if [tex]0^\circ <\theta<90^\circ[/tex] .
Step by step explanation:
Step 1:
Complementary angles are those two angles whose sum is equal to [tex]90^\circ[/tex] .
The given angles are [tex]\theta{\text{ and 90}}-\theta[/tex] .
Now find the sum of the given angles to check whether these angle are complementary or not.
[tex]\theta+90-\theta =90[/tex]
It can be seen that the sum of the angles is [tex]90^\circ[/tex] .
Therefore, [tex]\theta{\text{ and 90}}-\theta[/tex] are complementary angles.
Step 2:
We know that the sine of any acute angle is equal to the cosine of its complement angle.
Therefore, [tex]\sin\theta=\cos\left({90-\theta }\right)[/tex] as [tex]\theta{\text{ and 90}}-\theta[/tex] are complementary angles.
Now consider all the options.
Option A: No, because angles measuring [tex]\theta{\text{ and 90}}-\theta[/tex] are supplements.
The supplementary angles are those two angles whose sum is equal to [tex]180^\circ[/tex] .
As it has been proved that [tex]\theta{\text{ and 90}}-\theta[/tex] are complementary angles.
Thus, option A is not correct.
Option B: No, because angles measuring [tex]\theta{\text{ and 90}}-\theta[/tex] are complements.
It has been proved that [tex]\sin\theta=\cos\left({90-\theta}\right)[/tex] as [tex]\theta{\text{ and 90}}-\theta[/tex] are complementary angles.
Thus, option B is not correct.
Option C: Yes, because angles measuring [tex]\theta{\text{ and 90}}-\theta[/tex] are supplements.
This option is also not correct.
Option D: Yes, because angles measuring [tex]\theta{\text{ and 90}}-\theta[/tex] are complements.
It has been proved that [tex]\sin\theta=\cos\left({90-\theta}\right)[/tex] as [tex]\theta{\text{ and 90}}-\theta[/tex] are complementary angles.
Thus, option D is correct.
Learn more:
Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Trignometry
Keywords: sine function, cosine function, complementary angles, supplementary angles, measurement, angles, acute angle, complement, sum of the angles, supplements.
