Final answer:
To find the trigonometric functions of an acute angle θ in a right triangle, we use the opposite side (y), adjacent side (x), and hypotenuse (h). The sine (sin(θ)) is y/h, cosine (cos(θ)) is x/h, tangent (tan(θ)) is y/x, and cotangent (cot(θ)) is x/y. Specific measurements are needed to calculate these ratios.
Step-by-step explanation:
To find the exact values of the trigonometric functions of an acute angle θ in a right triangle, we can use the definitions based on the lengths of the sides of the triangle. Given an angle θ, the sine of θ (sin(θ)) is the ratio of the length of the side opposite the angle (y) to the length of the hypotenuse (h). The cosine of θ (cos(θ)) is the ratio of the length of the side adjacent to the angle (x) to the hypotenuse (h). The tangent of θ (tan(θ)) is the ratio of the opposite side (y) to the adjacent side (x). Finally, the cotangent of θ (cot(θ)) is the reciprocal of the tangent, which is the ratio of the adjacent side (x) to the opposite side (y).
To calculate these values, you would need the specific measurements of the sides of the triangle relative to angle θ. For example:
- sin(θ) = y/h
- cos(θ) = x/h
- tan(θ) = y/x
- cot(θ) = x/y
It is also important to note that these trigonometric functions are related through various identities, such as sin^2(θ) + cos^2(θ) = 1 and tan(θ) = sin(θ)/cos(θ). Additionally, the Pythagorean theorem can help determine the lengths of the sides if one side and the hypotenuse are known, as a² + b² = c².