find the exact values of the six trig functions of theta equals six pi divided by 8
6π/8 radians = 135°
135° is in quadrant II
The reference angle of 135° is 45°
45° is found in the 1-1-√2 right-angled triangle
In quadrant II , the sine and cosecant are positive
so
sin (6π/8) = 1/√2 , csc (6π/8) = √2
cos (6π/8) = -1/√2 , sec (6π/8) = -√2
tan (6π/8) = -1 , cot (6π/8) =-1
To find the exact values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle, you'll need to determine the values of the sides of a right triangle.
Given that theta equals 6π/8, we can simplify this fraction. Since π is a multiple of 180 degrees, you can convert it to degrees before proceeding.
6π/8 can be converted to degrees by using the following relationship: 1 radian = 180/π degrees.
6π/8 = (6/8) * (180/π) degrees
Simplifying, we get:
θ = 6π/8 = (3/4) * (180/π) degrees
θ ≈ (3/4) * 57.3 degrees ≈ 42.9 degrees
Now, we can use this angle to construct a right triangle and calculate the trigonometric ratios.
Step 1: Draw a right triangle with one angle of 42.9 degrees.
Step 2: Label the sides of the triangle: adjacent side (A), opposite side (O), and hypotenuse (H).
Step 3: Determine the lengths of the sides of the triangle. Since we don't have specific length values, we'll use variables.
Step 4: Calculate the trigonometric ratios using the definitions:
Sine (sin): sin(θ) = O/H
Cosine (cos): cos(θ) = A/H
Tangent (tan): tan(θ) = O/A
Cosecant (csc): csc(θ) = 1/sin(θ) = H/O
Secant (sec): sec(θ) = 1/cos(θ) = H/A
Cotangent (cot): cot(θ) = 1/tan(θ) = A/O
Substituting the values into the formulas, we have:
sin(θ) = O/H = O/1 = O
cos(θ) = A/H = A/1 = A
tan(θ) = O/A = O/A
csc(θ) = H/O = 1/O
sec(θ) = H/A = 1/A
cot(θ) = A/O = A
Given that θ = 42.9 degrees, the exact values of the six trigonometric functions are:
sin(θ) = O = ?
cos(θ) = A = ?
tan(θ) = O/A = ?
csc(θ) = 1/O = ?
sec(θ) = 1/A = ?
cot(θ) = A/O = ?
Since we don't have specific values for the lengths of the sides of the triangle, we can't determine the exact values of the trigonometric functions for θ.