Question

A point P(x,y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t. The point P is (-4/5, 3/5).

Answer 1

Answer:

sin t = $\frac{3}{5}$

cos t = $-\frac{4}{5}$

tan t = $-\frac{3}{4}$

csc t = $\frac{5}{3}$  

sec t = $-\frac{5}{4}$  

cot t = $-\frac{4}{3}$  

Step-by-step explanation:

In the unit circle:

• x-coordinate of a point on the circle represents cosine the angle between the + ve part of x-axis and the terminal side which joins the center of the circle and this point
• y-coordinate of a point on the circle represents sine the angle between + ve part of x-axis and the terminal side which joins the center of the circle and this point

In the attached figure

∵ t represents the angle between + ve part of x-axis

    and the terminal side drawn from the center of the circle to

    point P

∴ The coordinates of point P are (cos t , sin t)

∵ The coordinates of P are ( $-\frac{4}{5}$ , $\frac{3}{5}$ )

∴ sin t = $\frac{3}{5}$

∴ cos t = $-\frac{4}{5}$

∵ tan t = $\frac{sin(t)}{cos(t)}$

- Substitute the values of sin t and cos t

∴ tan t = $\frac{\frac{3}{5}}{-\frac{4}{5}}$

- Multiply up and down by 5 to simplify the fraction

∴ tan t = $-\frac{3}{4}$

∵ csc t = $\frac{1}{sin(t)}$

- Reciprocal the value of sin t

∴ csc t = $\frac{5}{3}$  

∵ sec t = $\frac{1}{cos(t)}$

- Reciprocal the value of cos t

∴ sec t = $-\frac{5}{4}$  

∵ cot t = $\frac{1}{tan(t)}$

- Reciprocal the value of tan t

∴ cot t = $-\frac{4}{3}$