Trigonometric ratios: csc, sec, and cot

Learning Objectives Define csc, sec, and cot as the reciprocals of sin, cos, and tan, respectively. Calculate the values of csc, sec, and cot for an angle in a right-angled triangle. Determine the exact values of csc, sec, and cot for special angles (e.g., π/6, π/4, π/3, π/2) using the unit circle. Find the values of all six trigonometric ratios given one ratio and the quadrant of the angle. Evaluate trigonometric expressions involving csc, sec, and cot. Apply the Pythagorean identities involving cot, sec, and csc to solve problems. Identify the angles for which csc, sec, and cot are undefined. You've mastered SOH CAH TOA, but what happens when you flip those famous ratios upside down? 🤔 Let's explore the powerful 'other three' trigonometric functions!...

TermDefinitionExample Reciprocal Trigonometric RatiosThree trigonometric ratios that are the multiplicative inverses (reciprocals) of the primary ratios (sine, cosine, tangent).Since sin(30°) = 1/2, its reciprocal ratio, csc(30°), is 1 / (1/2) = 2. 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 opposite side.If sin(θ) = 3/5, then csc(θ) = 5/3. In a triangle, if opposite = 3 and hypotenuse = 5, csc(θ) = hypotenuse/opposite = 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 adjacent side.If cos(θ) = 4/5, then sec(θ) = 5/4. In a triangle, if adjacent = 4 and hypotenuse = 5, sec(θ) = hypotenuse/ad...

Reciprocal Identities csc(θ) = 1/sin(θ) | sec(θ) = 1/cos(θ) | cot(θ) = 1/tan(θ) These fundamental identities define csc, sec, and cot in terms of sin, cos, and tan. Use them to convert between primary and reciprocal ratios. Remember that sin(θ), cos(θ), and tan(θ) cannot be zero for these to be defined. Quotient Identity for Cotangent cot(θ) = cos(θ)/sin(θ) Just as tan(θ) = sin(θ)/cos(θ), the cotangent is the ratio of cosine to sine. This is useful for simplifying expressions and proving other identities. Pythagorean Identities 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ) These are derived from the primary Pythagorean identity sin²(θ) + cos²(θ) = 1. They are essential for finding the value of one trig function when another is known, without needing to find t...