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Use the figures to find the exact value of each trigonometric function.

 

 Mar 8, 2021
 #1
avatar+498 
-1

\(\sin\theta = \frac{3}{5}\)

\(\cos \theta = \frac{4}{5}\)
 

Hypotenuse of the second triangle is: \(\sqrt{24^2+7^2} = 25\)

 

\(\sin \alpha = \frac{7}{25}\)

\(\tan \alpha = \frac{7}{24}\)

 

This means: 

\(\theta = \arcsin \frac{3}{5} ≈ 36.87\)

\(\alpha = \arcsin \frac{7}{25} ≈ 16.26\)

 

Now:
1. \(\sin 2\theta ≈ \sin 2(36.87) ≈ \sin73.74 = 0.96\)

2. \(\cos 2\theta ≈ \cos 2(36.87) ≈ \cos 73.74 = 0.28\)

3. \(\sin 2\alpha ≈ \sin 2(16.26) ≈ \sin 32.52 = 0.5376\)

4. \(\tan 2\alpha ≈ \tan 2(16.26) ≈ \tan 32.52 ≈ 0.6376\)

 Mar 8, 2021
 #2
avatar+31195 
+1

sin 2theta = 2 sin cos

                 = 2 ( 3/5)(4/5) = 24/25

 

cos 2theta = cos^2 - sin^2

                  = (4/5 )^2  - (3/5)^2   =  16/25 - 9/25 =  7/25

 

 

SImilar for alpha....use double angle identities .....           tan(2alpha) =  2 tan / (1- tan) .

 Mar 8, 2021

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