To evaluate the six trigonometric functions of an acute angle θ\thetaθ in a right triangle, we need to know the lengths of the sides of the triangle. Let's assume the lengths of the opposite side, adjacent side, and hypotenuse are given as opposite, adjacent, and hypotenuse, respectively. The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent, which can be calculated using these side lengths.
opposite
adjacent
hypotenuse
In a right triangle, we are given the side lengths: the opposite side is 3, the adjacent side is 4, and the hypotenuse is 5.
The sine of angle θ\thetaθ is calculated as the ratio of the length of the opposite side to the hypotenuse: sin(θ)=oppositehypotenuse=35=0.6000 \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} = 0.6000 sin(θ)=hypotenuseopposite=53=0.6000
The cosine of angle θ\thetaθ is calculated as the ratio of the length of the adjacent side to the hypotenuse: cos(θ)=adjacenthypotenuse=45=0.8000 \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} = 0.8000 cos(θ)=hypotenuseadjacent=54=0.8000
The tangent of angle θ\thetaθ is calculated as the ratio of the length of the opposite side to the adjacent side: tan(θ)=oppositeadjacent=34=0.7500 \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} = 0.7500 tan(θ)=adjacentopposite=43=0.7500
The cosecant of angle θ\thetaθ is the reciprocal of the sine: csc(θ)=1sin(θ)=53≈1.6667 \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{3} \approx 1.6667 csc(θ)=sin(θ)1=35≈1.6667
The secant of angle θ\thetaθ is the reciprocal of the cosine: sec(θ)=1cos(θ)=54=1.2500 \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{4} = 1.2500 sec(θ)=cos(θ)1=45=1.2500
The cotangent of angle θ\thetaθ is the reciprocal of the tangent: cot(θ)=1tan(θ)=43≈1.3333 \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{4}{3} \approx 1.3333 cot(θ)=tan(θ)1=34≈1.3333
sin(θ)=35,cos(θ)=45,tan(θ)=34,csc(θ)=53,sec(θ)=54,cot(θ)=43 \sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}, \quad \csc(\theta) = \frac{5}{3}, \quad \sec(\theta) = \frac{5}{4}, \quad \cot(\theta) = \frac{4}{3} sin(θ)=53,cos(θ)=54,tan(θ)=43,csc(θ)=35,sec(θ)=45,cot(θ)=34
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