tan ø = 5; Quad I
How do I find the other 5 function values?
since tanØ = opposite/adjacent or y/x
it is simple to draw a right angled triangle in the first quadrant where the base is 1 and the height is 5, making the hypotenuse √26
Now it is easy .....
sinØ = 5√26 , cscØ = √26/5
cosØ = 1/√26 , secØ = √26
tanØ = 5 , cotØ = 1/5
To find the other five trigonometric function values, we need to know the value of the angle ø in the first quadrant, and its tangent value. Since you provided the tangent value of ø, we can use that information to find the values of sine, cosine, cotangent, secant, and cosecant.
1. Sine (sin): To find sine, we divide the opposite side by the hypotenuse. In this case, we don't have a right triangle, so we need to create one. We know that the tangent of ø is equal to the ratio of the opposite side to the adjacent side. We can choose any convenient values for the opposite and adjacent sides, keeping in mind that we need to be in the first quadrant. Let's assume the opposite side is 5 and the adjacent side is 1. Using the Pythagorean theorem, we can find the hypotenuse.
Applying the Pythagorean theorem: h^2 = 5^2 + 1^2
Simplifying the equation: h^2 = 26
Taking the square root on both sides: h ≈ 5.099
Now we have a right triangle with the opposite side (5), adjacent side (1), and hypotenuse (approximately 5.099). Now we can find the sine value by dividing the opposite side by the hypotenuse.
sin ø = opposite/hypotenuse = 5/5.099 ≈ 0.981
2. Cosine (cos): To find cosine, we divide the adjacent side by the hypotenuse. Using the values from above,
cos ø = adjacent/hypotenuse = 1/5.099 ≈ 0.196
3. Cotangent (cot): To find cotangent, we calculate the reciprocal of the tangent.
cot ø = 1/tan ø = 1/5 ≈ 0.2
4. Secant (sec): To find secant, we calculate the reciprocal of cosine.
sec ø = 1/cos ø = 1/0.196 ≈ 5.102
5. Cosecant (csc): To find cosecant, we calculate the reciprocal of sine.
csc ø = 1/sin ø = 1/0.981 ≈ 1.019
Remember, these values are approximate since we rounded them to a reasonable number of decimal places.