Answer:
[tex]\sin(\theta)=\dfrac{3\sqrt{34}}{34}\\\\\sec(\theta)=\dfrac{\sqrt{34}}{5}\\\\\tan(\theta)=\dfrac{3}{5}[/tex]
Step-by-step explanation:
You want the sine, secant, and tangent of the angle θ in a right triangle with opposite side 3 and adjacent side 5.
The length of the hypotenuse is found using the Pythagorean theorem.
c² = a² +b² . . . . . . where a, b are the legs of the triangle
c² = 5² +3² = 25 +9 = 34
c = √34 . . . . . . . . the length of the hypotenuse
You are reminded how to find the trig functions by SOH CAH TOA. This is intended to help you remember ...
Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent
You need to remember also that the secant is the reciprocal of the cosine, so ...
Sec = Hypotenuse/Adjacent
Using the measures ...
these ratios are ...
[tex]\boxed{\begin{array}{l}\sin(\theta)=\dfrac{3\sqrt{34}}{34}\\\\\sec(\theta)=\dfrac{\sqrt{34}}{5}\\\\\tan(\theta)=\dfrac{3}{5}\end{array}}[/tex]
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Additional comment
The sine is O/H = 3/√34. The denominator is rationalized by multiplying this fraction by (√34)/(√34) to give the value shown above.