Supposed Sin Theta=3/10 And Sec Theta <0. Find The Values
Trigonometry involves finding the tangent of a given point, as shown in the equation tan(theta)=3√919122. A free math problem solver provides step-by-step explanations for various subjects, including algebra, geometry, trigonometry, calculus, and statistics. Opposite numbers, such as 5 and -5, are equidistant from zero on a number line in opposite directions.
To find the values of sin(2θ) and cos(2θ), we can use the relationship between the double angle formulas and the given information about the trigonometric functions.
First, we know that sec(θ) < 0. The secant function is negative in the second and third quadrants. Given that the sine function is positive (sin(θ) = 3/10), we can conclude that the angle θ lies in the second quadrant.
Since sin(θ) = 3/10, we can determine the third side of the triangle using the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1. This gives us cos(θ) = -√(1 - sin^2(θ)) = -√(1 - (3/10)^2) = -√(1 - 9/100) = -√(91/100) = -(√91)/10.
Now, we can find sin(2θ) and cos(2θ) using the double angle formulas: sin(2θ) = 2*sin(θ)cos(θ) and cos(2θ) = 2cos^2(θ) - 1.
Plugging in the known values: sin(2θ) = 2*(3/10)(-√91/10) ≈ -0.564 cos(2θ) = 2(-√91/10)^2 - 1 ≈ -0.672
So, sin(2θ) ≈ -0.564 and cos(2θ) ≈ -0.672.
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