Find the exact value of each trigonometric function.
sin(θ)= 3/5
cos(θ)= 4/5
tan(θ)= 3/4
1. cos(2θ)=
2. tan(2θ)=
3. cot(2θ)=
4. sec(2θ )=
5. csc(2θ)=
You are given all 3 of the major trig ratios of Ø
(btw, you are dealing with the 3-4-5 Pythagorean triangle, the simplest of such triangles)
You must memorize or have at hand from your text or notes, the following:
sin (2Ø) = 2 sinØcosØ
cos(2Ø) = cos^2 Ø - sin^2 Ø
All the other double angle trig ratios can be easily derived from those two
I will do one of the, say #3
cot 2Ø = cos 2Ø/sin 2Ø
= (cos^2 Ø - sin^2 Ø)/(2sinØcosØ)
= (16/25 - 9/25)/(2(3/5)(4/5) )
= (7/25) / (24/25)
= (7/25)(25/24) = 7/24
let me know what you get for the others.
Hey, you have a textbook I suspect that shows the trig formulas you need
for example cos (2 T) = 2 cos^2 (T) - 1
or = cos^2(T) - sin^2 (T)
I am not going to do the arithmetic for you
To find the values of the trigonometric functions for 2θ, we can use the double-angle trigonometric identities. These identities allow us to express the trigonometric functions of 2θ in terms of the trigonometric functions of θ.
1. cos(2θ) = 2cos^2(θ) - 1
To find cos(2θ), we need to square the given value of cos(θ) and subtract 1:
cos^2(θ) = (4/5)^2 = 16/25
cos(2θ) = 2(16/25) - 1 = 32/25 - 25/25 = 7/25
2. tan(2θ) = 2tan(θ) / (1 - tan^2(θ))
To find tan(2θ), we need to substitute the values of tan(θ) into the equation:
tan(θ) = 3/4
tan^2(θ) = (3/4)^2 = 9/16
tan(2θ) = 2*(3/4) / (1 - 9/16) = 6/4 / (16/16 - 9/16) = 3/2 / 7/16 = 48/14 = 24/7
3. cot(2θ) = 1 / tan(2θ)
To find cot(2θ), we need to find the reciprocal of tan(2θ) calculated in step 2:
cot(2θ) = 1 / (24/7) = 7/24
4. sec(2θ) = 1 / cos(2θ)
To find sec(2θ), we need to find the reciprocal of cos(2θ) calculated in step 1:
sec(2θ) = 1 / (7/25) = 25/7
5. csc(2θ) = 1 / sin(2θ)
To find csc(2θ), we need to find the reciprocal of sin(2θ). We can use the double-angle formula for sine: sin(2θ) = 2sin(θ)cos(θ)
sin(θ) = 3/5
sin(2θ) = 2 * (3/5) * (4/5) = 24/25
csc(2θ) = 1 / (24/25) = 25/24
So, the exact values of the trigonometric functions are:
1. cos(2θ) = 7/25
2. tan(2θ) = 24/7
3. cot(2θ) = 7/24
4. sec(2θ) = 25/7
5. csc(2θ) = 25/24