Answer :
To determine whether there are values of \( t \) such that \( \sin t = 0.35 \) and \( \cos t = 0.6 \), we need to check if these values satisfy the trigonometric identity:
[tex]\[ \sin^2 t + \cos^2 t = 1 \][/tex]
Let's substitute the given values into the identity:
[tex]\[ (0.35)^2 + (0.6)^2 \][/tex]
First, we calculate the squares:
[tex]\[ (0.35)^2 = 0.1225 \][/tex]
[tex]\[ (0.6)^2 = 0.36 \][/tex]
Next, we add these results together:
[tex]\[ 0.1225 + 0.36 = 0.4825 \][/tex]
This sum does not equal 1, given that:
[tex]\[ 0.4825 \neq 1 \][/tex]
Since the fundamental trigonometric identity \( \sin^2 t + \cos^2 t \) must equal 1 for all values of \( t \), and this equation does not hold with the given values, it concludes that there are no values of \( t \) such that both \( \sin t = 0.35 \) and \( \cos t = 0.6 \) at the same time.
Therefore, the correct answer is:
B. False
[tex]\[ \sin^2 t + \cos^2 t = 1 \][/tex]
Let's substitute the given values into the identity:
[tex]\[ (0.35)^2 + (0.6)^2 \][/tex]
First, we calculate the squares:
[tex]\[ (0.35)^2 = 0.1225 \][/tex]
[tex]\[ (0.6)^2 = 0.36 \][/tex]
Next, we add these results together:
[tex]\[ 0.1225 + 0.36 = 0.4825 \][/tex]
This sum does not equal 1, given that:
[tex]\[ 0.4825 \neq 1 \][/tex]
Since the fundamental trigonometric identity \( \sin^2 t + \cos^2 t \) must equal 1 for all values of \( t \), and this equation does not hold with the given values, it concludes that there are no values of \( t \) such that both \( \sin t = 0.35 \) and \( \cos t = 0.6 \) at the same time.
Therefore, the correct answer is:
B. False
