Answer:
- y = √119 ≈ 10.909
- θ ≈ 65.376°
- α ≈ 24.624°
- sin(θ) = (√119)/12 ≈ 0.909059
- cos(θ) = 5/12 ≈ 0.416667
- tan(θ) = (√119)/5 ≈ 2.181742
- csc(θ) = (12√119)/119 ≈ 1.100038
- sec(θ) = 2.4
- cot(θ) = (5√119)/119 ≈ 0.458349
Step-by-step explanation:
You can use the Law of Cosines after you find y or θ. Using the given information, you can apply the Pythagorean theorem, the Law of Sines, or the definition of the cosine trig function.
The mnemonic SOH CAH TOA is intended to remind you of the relations between sides and angles in a right triangle. It tells you ...
Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Find y
The Pythagorean theorem relates the measures of the sides. It tells you ...
y² +5² = 12²
y = √(144 -25) = √119
Find angles
The trig relations above tell you ...
sin(α) = 5/12 ⇒ α = arcsin(5/12) ≈ 24.624°
cos(θ) = 5/12 ⇒ θ = arccos(5/12) ≈ 65.376°
Of course, once you find one of the angles, you can find the other. The sum of the two acute angles in a right triangle is 90°.
Find trig functions
Using the definitions above for the trig functions, you have ...
sin(θ) = y/12 = (√119)/12
cos(θ) = 5/12
tan(θ) = y/5 = (√119)/5
Using trig identities, you have ...
csc(θ) = 1/sin(θ) = 12/√119 = (12√119)/119
sec(θ) = 1/cos(θ) = 12/5 = 2.4
cot(θ) = 1/tan(θ) = 5/√119 = (5√119)/119
_____
Additional comment
The attached calculator screen shot shows you the angle θ on the first line, and the value of y on the second line. The csc, sec, and cot are shown on the third line. You will notice the angle mode (DEG) is seen in the lower left corner. If the calculator angle mode is set to radians, you will see the angles as relatively small radian values: θ ≈ 1.141, α ≈ 0.4298.
We have shown both exact values and decimal values you can round to the desired precision.
