Given cos(t)=-7/25 for pi/2 < t < pi, find the value of sin (t)

Question

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Answer 1

Since we have the value for cos(t), we can use the trigonometric identity cos²(t)+sin²(t)=1 to find the value for sin(t) by substituting in the value of cos(t). Namely,

(cos(t))² + (sin(t))² = 1

(-7/25)² + (sin(t))² = 1, substituted cos(t) = -7/25

(sin(t))² = 25²/25² - 7²/25²

(sin(t))² = (25²-7²) / 25²

(sin(t))² = (25-7)(25+7) / 25², we factored the numerator as a difference of squares

(sin(t))² = (18)(32) / 25²,

Moving factors around in the numerator for a clearer expression gives us

(sin(t))² = (9*2)(2⁵) / 25² = 3²*2⁶ / 25²,

Then we take the square root on both sides,

sin(t) = ±(3²*2⁶ / 25²)^1/2 = ±(3*2³ / 25) = ± 24/25

Since we are told that pi/2 < t < pi, we know that t is in the second quadrant, and the sine function is positive in the second quadrant, therefore

sin(t) = 24/25

We can verify our answer by substituting both values in the trigonometric identity used above,

(-7/25)² + (24/25)² = 49/625 + 576/625 = 625/625 = 1