Six-Trig Ratio│Please Help!

Question

Given the point (8, -3√5) determine the six-trig ratio. Determine the two complementary angles. Label the angles Q, R, S, with R being the right angle.

Answer 1

Given Point

The given point is (8, -3√5).

Calculating the Radius

The radius r is calculated using the formula:

r = √(x² + y²)

Substituting the values:

r = √(8² + (-3√5)²)

r = √(64 + 45) = √(109)

Trigonometric Ratios

1. Sine (sin):

  sin(Q) = y/r = (-3√5)/√(109)

2. Cosine (cos):

  cos(Q) = x/r = 8/√(109)

3. Tangent (tan):

  tan(Q) = y/x = (-3√5)/8

4. Cosecant (csc):

  csc(Q) = r/y = √(109)/(-3√5) = -√(109)/(3√5)

5. Secant (sec):

  sec(Q) = r/x = √(109)/8

6. Cotangent (cot):

  cot(Q) = x/y = 8/(-3√5) = -8/(3√5)

Complementary Angles

In a right triangle where R is the right angle, the angles Q and S are complementary.

Let S be the angle complementary to Q:

S = 90° - Q

Summary of Trigonometric Ratios

sin(Q) = (-3√5)/√(109)

cos(Q) = 8/√(109)

tan(Q) = (-3√5)/8

csc(Q) = -√(109)/(3√5)

sec(Q) = √(109)/8

cot(Q) = -8/(3√5)

Angles

Q is the angle formed by the radius with the positive x-axis.

R = 90° (right angle).

S = 90° - Q (complementary to Q).

Hope this helps! :)