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omeli [17]
3 years ago
7

What is the half-angle of tangent 22.5 (degrees)?

Mathematics
2 answers:
Sergio039 [100]3 years ago
6 0

Find tan (22.5)

Answer: #-1 + sqrt2#

Explanation:

Call tan (22.5) = tan t --> tan 2t = tan 45 = 1

Use trig identity: # tan 2t = (2tan t)/(1 - tan^2 t)# (1)

#tan 2t = 1 = (2tan t)/(1 - tan^2 t)# -->
--> #tan^2 t + 2(tan t) - 1 = 0#
Solve this quadratic equation for tan t.

#D = d^2 = b^2 - 4ac = 4 + 4 = 8# --> #d = +- 2sqrt2#
There are 2 real roots:
tan t = -b/2a +- d/2a = -2/1 + 2sqrt2/2 = - 1 +- sqrt2
Answer:
#tan t = tan (22.5) = - 1 +- sqrt2#
Since tan 22.5 is positive, then take the positive answer:
tan (22.5) = - 1 + sqrt2


irina [24]3 years ago
6 0
Since <span>22.522.5</span> is not an angle where the values of the six trigonometric functions are known, try using half-angle identities.<span>22.522.5</span> is not an exact angleFirst, rewrite the angle as the product of <span><span>12</span><span>12</span></span> and an angle where the values of the six trigonometric functions are known. In this case, <span>22.522.5</span> can be rewritten as <span><span><span>(<span>12</span>)</span>⋅45</span><span><span>12</span>⋅45</span></span>.<span><span>tan<span>(<span><span>(<span>12</span>)</span>⋅45</span>)</span></span><span>tan<span><span>12</span>⋅45</span></span></span>Use the half-angle identity for tangent to simplify the expression. The formula states that <span><span><span><span><span>ta</span>n</span><span>(<span>θ2</span>)</span></span>=<span><span>sin<span>(θ)</span></span><span>1+<span>cos<span>(θ)</span></span></span></span></span><span><span><span><span>ta</span>n</span><span>θ2</span></span>=<span><span>sinθ</span><span>1+<span>cosθ</span></span></span></span></span>.<span><span><span>sin<span>(45)</span></span><span>1+<span>cos<span>(45)</span></span></span></span><span><span>sin45</span><span>1+<span>cos45</span></span></span></span>Simplify the result.
<span><span>√2</span><span>−<span>1</span></span></span>
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