Given that tanθ = 2 root 10 over 9 and cscθ < 0 , find the exact value of cos(θ − pie over 4) .
I solved for the other side and got 11. After this idk what to do I was thinking using
cos(a-b)=cos(a)cos(b)-sin(a)sin(b) but idk how to witht the pie/4 someone please help
tanØ = 2√10/9
recall tanØ = y/x in the corresponding right-angled triangle
we are also given that cscØ is negative, so Ø must be in III
then y = -2√10 and x = -9
r^2 = 40 + 81 =121
r = 11
x = -9, y = -2√10, r = 11
sinØ = -2√10/11, cosØ = -9/11
cos(Ø - π/2)
= cosØcos π/2 + sinØsin π/2
= (-9/11)(0) + (-2√10/11)(1)
= -2√10/11
π/2 radians = 90°
You should know the trig functions of sine, cosine, and tangents of the main angles .
30°, 60°, 90° from the standard right-angled triangle with corresponding sides 1 : √3 : 2
and the 45 - 45 - 90 triangle with sides
1 : 1 : √2
trig functions of 0, 90, 180 270 and 360 you should know by looking at their curves.
well, the sides could be 1, (2/9)sqrt10 and hypotenuse 11/9
or
9, 2 sqrt 10, 11 with 2 sqrt 10 opposite θ
we know sin θ <0 so θ in quadrant 3 or 4
since tan is +, must be quadrant 3
now actually
cos(a-b)=cosa cosb + sina sinb
cos a = 9/11
cos b = 1/sqrt 2
sin a = -(2/11) sqrt10
sin b = 1/sqrt2
so
9/(11 sqrt 2) - (2/11)sqrt 5
sin a = -2 sqrt 10
but she said pi/4 :)
oh, cos also - in quadrant 3
cos a = -9/11
To find the exact value of cos(θ − π/4), we can use the angle sum formula for cosine:
cos(θ − π/4) = cos(θ)cos(π/4) + sin(θ)sin(π/4)
We are given that tan(θ) = 2√10/9.
Using the definitions of tan and sin, we have:
sin(θ) = tan(θ) / √(1 + tan^2(θ))
= (2√10/9) / √(1 + (2√10/9)^2)
Simplifying this expression, we get:
sin(θ) = (2√10/9) / √(1 + 40/81)
= (2√10/9) / √(121/81)
= (2√10/9) / (11/9)
= 2√10 / 11
Now let's find cos(θ):
cos^2(θ) = 1 - sin^2(θ)
= 1 - (2√10/11)^2
= 1 - (40/121)
= (121 - 40) / 121
= 81 / 121
Since csc(θ) < 0, we know that sin(θ) is negative. From our previous calculation, sin(θ) = 2√10 / 11, which is positive. Therefore, θ must be in the second quadrant, where cosine is negative. So we take the negative square root:
cos(θ) = -√(81/121)
= -9/11
Now let's find cos(π/4):
cos(π/4) = 1/√2
= √2/2
Now we can substitute these values into the angle sum formula:
cos(θ − π/4) = cos(θ)cos(π/4) + sin(θ)sin(π/4)
= (-9/11)(√2/2) + (2√10/11)(1/√2)
= -9√2 / (11√2) + (2√10/√2) / 11
= -9√2 / 11√2 + (2√10/√2) / 11
= -9/11 + 2√10/11
= (2√10 - 9) / 11
So the exact value of cos(θ − π/4) is (2√10 - 9) / 11.