Trigonometric ratios: csc, sec, and cot

Learning Objectives Define cosecant (csc), secant (sec), and cotangent (cot) as reciprocals of the primary trigonometric ratios. Express csc, sec, and cot in terms of the opposite, adjacent, and hypotenuse sides of a right-angled triangle. Calculate the exact values of all six trigonometric ratios for a given right-angled triangle. Determine the values of the other five trigonometric ratios when one ratio is known. Use a calculator to approximate the values of csc, sec, and cot for a given angle. Solve for missing side lengths in right-angled triangles using csc, sec, and cot. You've mastered sine, cosine, and tangent, but what happens when we flip them upside down? 🤔 Let's explore the 'reciprocal' family of trigonometric ratios! This tutorial introduce...

TermDefinitionExample ReciprocalThe multiplicative inverse of a number. To find the reciprocal of a fraction, you flip it upside down.The reciprocal of 2/3 is 3/2. The reciprocal of 5 (which is 5/1) is 1/5. Cosecant (csc)The reciprocal of the sine ratio. In a right-angled triangle, it is the ratio of the length of the hypotenuse to the length of the side opposite the angle.If sin(θ) = 3/5, then csc(θ) = 5/3. Secant (sec)The reciprocal of the cosine ratio. In a right-angled triangle, it is the ratio of the length of the hypotenuse to the length of the side adjacent to the angle.If cos(θ) = 4/5, then sec(θ) = 5/4. Cotangent (cot)The reciprocal of the tangent ratio. In a right-angled triangle, it is the ratio of the length of the side adjacent to the angle to the length of the side opposite...

Reciprocal Identities csc(θ) = 1 / sin(θ) sec(θ) = 1 / cos(θ) cot(θ) = 1 / tan(θ) These are the fundamental definitions of the reciprocal ratios. Use these to find the value of a reciprocal ratio if you already know the value of its primary partner (sin, cos, or tan). Right-Triangle Definitions csc(θ) = Hypotenuse / Opposite sec(θ) = Hypotenuse / Adjacent cot(θ) = Adjacent / Opposite These formulas allow you to calculate the reciprocal ratios directly from the side lengths of a right-angled triangle, without first finding sin, cos, or tan. Quotient Identity for Cotangent cot(θ) = cos(θ) / sin(θ) Since tan(θ) = sin(θ) / cos(θ) and cot(θ) is its reciprocal, we can express cot(θ) as the ratio of cos(θ) to sin(θ). This is useful in proofs and simplifying expressi...