What is the value of tan (Arc cos 15 seventeenths)?
1. 8/15
2. 8/17
3. 15/8
4. 17/8
Can you guys explain what the Arc means?
suppose you have:
cos 60° = 1/2 , we know that
then arc cos(1/2) = 60°
other notations are: cos^-1 (1/2) = 60°
So, arc cos and cos are inverse operations.
On your calculator you will notice that each trig function
has its inverse after pressing 2ndF
e.g. enter
sin 45° , you will get .707106...
now press 2ndF, sin, and you will get back your 45°
so, to your question,
look at the Arc cos 15 seventeenths or, I assume you meant: arc cos (15/17)
This means you are looking for an angle Ø so that cosØ = 15/17
sketch a right-angled triangle with base angle Ø, hypotenuse of 17, and adjacent side of 15
by Pythagoras: 15^2 + y^2 = 17^2
y^2 = 64
y = √64 = 8
then tan (Arc cos (15/17)) = tanØ = opposite/adjacent = 8/15 or .5333...
Verification on my calculator
15/17
=
2ndF cos
=
tan
=
I got .53333..
Of course! In mathematics, "Arc" is short for arcus, which is Latin for "arc." In trigonometry, arc refers to the inverse function of a trigonometric ratio. For example, arc sine (Arc sin or asin) is the inverse function of sine.
In this question, we are asked to find the value of tan(Arc cos(15/17)).
To solve this, we can follow these steps:
1. Start with the expression tan(Arc cos(15/17)).
2. Consider the identity: cos²θ + sin²θ = 1, which means the square of the cosine of an angle plus the square of the sine of the same angle equals 1.
3. Rearrange the equation from step 2 to isolate the square of the sine: sin²θ = 1 - cos²θ.
4. Take the square root of both sides of the equation from step 3: sinθ = √(1 - cos²θ).
5. Since we are dealing with the cosine function, we can replace cos²θ with (15/17)², as given in the question.
6. Calculate the value of sinθ by substituting (15/17)² into the formula obtained in step 4.
7. Use the definition of the tangent function: tanθ = sinθ/cosθ.
8. Replace sinθ with the value obtained in step 6, and cosθ with (15/17).
9. Simplify the expression to find the value of tan(Arc cos(15/17)).
10. Compare the result obtained to the given options (1, 2, 3, or 4) to determine the correct answer.
So, by following these steps, you should be able to calculate the value of tan(Arc cos(15/17)) and determine the correct option.