Final answer:
To find the remaining trigonometric functions when sinθ is known, use trigonometric identities and the Pythagorean Theorem in a right triangle. Cosine can be found directly, and other functions like tangent or secant can be calculated based on sine and cosine values.
Step-by-step explanation:
To find the exact value of the remaining trigonometric functions given sinθ, one should use the fundamental trigonometric identities and relations in a right triangle. For instance, they know that sinθ = Ay/A, and assuming the right triangle with hypotenuse A, opposite side Ay, and adjacent side Ax, the Pythagorean Theorem can be applied as Ax² + Ay² = A².
Then, the cosine function can be determined using cosθ = Ax/A. If trigonometric identities like the Law of Sines or the Law of Cosines are needed for non-right triangles, they can also be applied accordingly.
When dealing with various trigonometric functions such as tangent, cotangent, secant, and cosecant, one can use the definitions based on sine and cosine (tanθ = sinθ/cosθ, cotθ = 1/tanθ, secθ = 1/cosθ, and cosecθ = 1/sinθ) to find their values. Trigonometric formulae such as double angle formulas (sin 2θ, cos 2θ) and sum-to-product identities (sin a + sin β, cos a + cos β) can provide additional relationships between functions.