Answer:
sin(Ф) = 4(√2)/9 ; cos(Ф) = 7/8 ; tan(Ф) = 4(√2)/7 ; csc(Ф) = (9√2)/8 ; sec(Ф) = 8/7 ; tan(Ф) = (7√2)/8.
Step-by-step explanation:
It would first be wise to solve for the hypotenuse side, or the longest side of the triangle.
We can use the Pythagorean Theorem: a² + b² = c²
Since we have both of the shorter sides, which are 14 and 8√2, we can solve for c.
14² + (8√2)² = c²
c = √324 = 18
We can then find and solve the 6 trig identities - sine, cosine, tangent, cosecant, secant, and cotangent - using the acronym "SOH CAH TOA" and other properties.
Starting with the most basic of the three: sine, cosine, and tangent:
1: SOH = Sine Opposite over Hypotenuse
Relative to theta, the opposite side has a value of 8√2 and the hypotenuse has a value of 18. Following the acronym, sine(Ф) = (8√2)/18, and simplifying that would leave you with (4√2)/9.
2: CAH = Cosine Adjacent over Hypotenuse
Relative to theta, the adjacent side has a value of 14 and the hypotenuse has a value of 18. Following the acronym, cosine(Ф) = 14/18, and simplifying that would leave you with 7/8.
3: TOA= Tangent Opposite over Adjacent
Relative to theta, the opposite side has a value of 8√2 and the hypotenuse has a value of 14. Following the acronym, tangent(Ф) = (8√2)/14, and simplifying that would leave you with (4√2)/7.
Next, we must note that, by definition, secant is the inverse of cosine, cosecant is the inverse of sine, and cotangent is the inverse of tangent. What that means is that you flip the numerator and the denominator of a fraction, or put a value to the power of -1.
Therefore:
1. Cosecant = 9/(4√2), and simplifying that leaves you with (9√2)/8.
2. Secant = 8/7.
3. Cotangent = 7/(4√2), and simplifying that leaves you with (7√2)/8.
Hope this helps!
