Question

what is the value of X in the angle please tell fast as you can ​

Answer 1

Step-by-step explanation:

Define AD to be of length 1.

Use the sine rule on ACD

⇒AC=sin(80)sin(40)=2cos(40)

Triangle ABD is isosceles so AB=AD=1

Now consider triangle ABC. We know sides AB and AC and we know the angle between them. We can find a missing angle for this triangle by the formula:

tan(C)=sin(A)bc−cos(A)

(I don't recall seeing this formula before so I just worked it out. I think it's true for every triangle. Shout if you disagree.)

In this case:

tan(x)=sin(20)2cos(40)1−cos(20)

tan(x)=sin(20)4cos2(20)−cos(20)−2

tan(x)=sin(20)3–√sin(20) (*)

tan(x)=13–√

So x = 30 degrees

* Here is a proof that

4cos2(20)−cos(20)−2=3–√sin(20)

Define s=sin(20) and c=cos(20).

Using the triple angle formulas:

12=cos(60)=4c3−3c

3√2=sin(60)=−4s3+3s

Multiply the first by c and the second by s and add:

c2+3√s2=4c4−3c2−4s4+3s2

=4(c2−s2)(c2+s2)−3(c2−s2)

=(c2−s2)(4−3)

=(2c2−1)

So

c2+3√s2=2c2−1

4c2−c−2=3–√s

Answer 2

Answer:

70

Step-by-step explanation:

40+75+x=180      (angle sum property)

110+x=180

x=180-110

x=70