The terminal point
P(x, y)
determined by a real number t is given. Find sin(t), cos(t), and tan(t).

−
3
5
, −
4
5

sin(t) =

cos(t) =

tan(t) =

We are given the terminal point \( P(x, y) = \left( -\frac{3}{5}, -\frac{4}{5} \right) \), and we need to find \( \sin(t) \), \( \cos(t) \), and \( \tan(t) \).

### 1. Finding \( \sin(t) \)
From the point \( P(x, y) \), we know that:
- \( x = -\frac{3}{5} \)
- \( y = -\frac{4}{5} \)

The sine of the angle \( t \) is given by:
\[
\sin(t) = \frac{y}{r}
\]
where \( r \) is the radius or the distance from the origin to the point \( P(x, y) \).

We first calculate \( r \):
\[
r = \sqrt{x^2 + y^2} = \sqrt{\left( -\frac{3}{5} \right)^2 + \left( -\frac{4}{5} \right)^2} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{\frac{25}{25}} = 1
\]

Now, \( r = 1 \), so:
\[
\sin(t) = \frac{y}{r} = \frac{-\frac{4}{5}}{1} = -\frac{4}{5}
\]

### 2. Finding \( \cos(t) \)
The cosine of the angle \( t \) is given by:
\[
\cos(t) = \frac{x}{r}
\]

Since \( r = 1 \), we have:
\[
\cos(t) = \frac{-\frac{3}{5}}{1} = -\frac{3}{5}
\]

### 3. Finding \( \tan(t) \)
The tangent of the angle \( t \) is given by:
\[
\tan(t) = \frac{\sin(t)}{\cos(t)} = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}
\]

### Final answers:
\[
\sin(t) = -\frac{4}{5}, \quad \cos(t) = -\frac{3}{5}, \quad \tan(t) = \frac{4}{3}
\]

Would you like more details or have any questions?

### Related Questions:
1. What is the value of \( \cot(t) \)?
2. How can we determine the quadrant in which \( t \) lies?
3. What is the secant function, and how can we calculate \( \sec(t) \)?
4. What is the cosecant function, and how can we calculate \( \csc(t) \)?
5. How do these values of \( \sin(t) \), \( \cos(t) \), and \( \tan(t) \) help us in solving trigonometric equations?

### Tip:
Always remember that \( r \), the distance from the origin to the point, is positive and is calculated using the Pythagorean theorem.

The terminal point P(x, y) determined by a real number t is given. Find sin(t), cos(t), and tan(t). −3/5, −4/5.

Learn how to find sin(t), cos(t), and tan(t) from the terminal point P(-3/5, -4/5). This trigonometry problem uses key concepts such as the unit circle and the Pythagorean theorem. Suitable for high school students (Grades 9-12).

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