Using a double angle formula how would you find the exact value of this expression?
Cos²105° - sin²105°
cos^2(105) - sin^2(105)
= cos 2(105)
= cos 210
= cos (180 + 30)
= cos180cos30 - sin30sin180
= -1(√3/2) - (1/2)(0)
= -√3/2
Hey thanks can you point me in the direction to solving this problem
2tan(5pi/12)/1-tan²(5pi/12)
Did you recoginize that your expression matches the formula for tan 2A ?
tan 2A = 2tanA/(1-tan^2 A)
so
2tan(5pi/12)/1-tan²(5pi/12)
= tan(5pi/6) or tan 150º
tan 150
= -tan 30 or - tan(pi/6)
= -1/√3
To find the exact value of the expression cos²105° - sin²105° using a double angle formula, we can use the formula for cos(2θ) and sin(2θ), where θ is the angle 105°.
The double angle formula for cosine (cos(2θ)) is given by:
cos(2θ) = cos²θ - sin²θ
Similarly, the double angle formula for sine (sin(2θ)) is given by:
sin(2θ) = 2sinθcosθ
Let's start by finding the values of cos²105° and sin²105°:
cos²105° = (cos210°)²
sin²105° = (sin210°)²
To express these values using the double angle formulas, we need to find double the angle 105°, which is 210°.
Using the formulas mentioned earlier, we can write:
cos²105° = cos²(2 * 52.5°) = cos²(105°)
sin²105° = sin²(2 * 52.5°) = sin²(105°)
Now, we can substitute these values into the double angle formulas:
cos²105° - sin²105° = cos(2 * 105°) - sin(2 * 105°)
cos(2 * 105°) can be written as cos(210°), and sin(2 * 105°) can be written as sin(210°).
Finally, we substitute these values into the expression:
cos²105° - sin²105° = cos(210°) - sin(210°)
To find the exact numerical value of cos(210°) and sin(210°), we need to evaluate them using a calculator or refer to the trigonometric values table.
After obtaining the values of cos(210°) and sin(210°), you can subtract them to find the exact value of the expression cos²105° - sin²105°.