Trigonometry 1 with The Math Sorcerer

Explore how to define and compute the six trigonometric functions—sine, cosine, tangent, and their reciprocals—using a point on the x y plane, r, and the Pythagorean theorem.

Explore how the signs of sine, cosine, and tangent vary by quadrant, using the 'all students take calculus' mnemonic and reciprocal relationships for cosecant, secant, and cotangent.

Explain the ranges of sine and cosine, and tangent and cotangent. Note that sine and cosine lie between -1 and 1, and tangent and cotangent cover all real numbers.

Learn the pythagorean identities, including sine squared plus cosine squared equals one, and derive tangent and cotangent forms; apply to find cosine theta from cosecant in quadrant three.

Identify the point (0,5) on the terminal side of theta, calculate r = 5, and determine trig values: sine 1, cosine 0, tangent undefined, cosecant 1, secant undefined, cotangent 0.

Compute all six trig function values for the angle with terminal point (5, -12) in quadrant iv by using r = 13 and sine, cosine, tangent formulas and their reciprocals.

Compute r = sqrt((-12)^2+(-5)^2) = 13, then sin = -5/13, cos = -12/13, tan = 5/12, with csc = -13/5, sec = -13/12, cot = 12/5.

Compute the six trig function values for theta from the coordinates (-4, -3) using r = sqrt(x^2 + y^2) = 5, yielding sines, cosines, tangents, and reciprocals.

Determine r from the point (-3,4) on theta's terminal side, then compute sin, cos, tan as 4/5, -3/5, -4/3, and the reciprocals secant -5/3, cosecant 5/4, cotangent -3/4.

Determine theta on the line sqrt(3)x + y = 0 with x ≤ 0, compute r = 2, and derive sin, cos, tan and their reciprocals for six trig values.

Choose a positive point on the line 3x+5y=0, take x=5, y=-3, compute r = sqrt(34), then find sine, cosine, tangent and their reciprocals.

determine sine, cosine, tangent, and their reciprocals for theta on the terminal side of the line 2x+y=0 with x≥0 by using the point (1,-2) and r=√5.

learn how to find secant theta when cosine theta is two thirds by using the reciprocal, yielding secant theta equals three over two.

In quadrant iii, cos theta = -3/5, so x = -3 and y = -4; sin = -4/5, tan = 4/3, csc = -5/4, sec = -5/3, cot = 3/4.

Derive cos theta in terms of sin theta from the identity cos^2 theta + sin^2 theta = 1, solving for cos theta and taking the positive root for acute angles.

Apply 1 + cot^2 theta = csc^2 theta to express sine in terms of cotangent theta, then take the square root and use quadrant iii to select the negative sign.

Define the trigonometric functions from a right triangle using adjacent, opposite, and hypotenuse. Use sine, cosine, and tangent and their reciprocals cosecant, secant, cotangent, via soh cah toa.

Learn a simple memorization trick for trig values by listing angles 0, 30, 45, 60, 90 and using the sqrt(n)/2 pattern to derive sine, cosine, and tangent using reference angles.

Learn cofunction identities for sine, cosine, secant, cosecant, tangent, and cotangent with theta in degrees and radians. Examples convert secant 39° to cosecant 51° and tan π/3 to cotangent π/6.

Learn to express a trig function in terms of its cofunction, using degrees or radians, by applying 90-degree or pi over two complements to cotangent, sine, and cosine.

Identify reference angles, defined as the positive acute angle between theta's terminal side and the x axis (theta prime). Use it to determine trig values by quadrant.

Find reference angles by measuring the acute angle between the terminal side and the x axis. Use examples with 240, 340, and -110 degrees to identify the reference angle.

Explore the unit circle with origin as center, where cosine is x and sine is y. Use it for quadrennial angles and points, and practice problems to see patterns.

This lecture uses the unit circle to compute trig functions at pi, showing cos pi = -1, sin pi = 0, and that secant, cosecant, and cotangent are undefined.

Find the reference angle for four pi over three and calculate the trig values for pi over three. Use quadrant three signs to determine sine, cosine, tangent, and reciprocals.

Identify five pi over three on the unit circle, find the reference angle pi over three, compute the basic values, then apply signs to get all trig functions.

Identify the reference angle pi over six and the quadrant four signs to compute sine, cosine, and tangent for 11 pi over six. Then derive csc, sec, and cot.

Master the trig function values for 30 degrees using a 30-60-90 triangle, memorizing sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent.

Derive the trig function values for 45 degrees using a 45–45–90 triangle and sohcahtoa, revealing sine and cosine as 1/√2, tangent as 1, and reciprocals csc and sec as √2.

Find all trig function values for 60 degrees using a 30-60-90 right triangle and the opposite, adjacent, and hypotenuse relationships. Memorize this triangle to quickly do 30 degrees as well.

Discover how to compute the cosine of zero using the unit circle, where cosine is the x-coordinate and the relevant point is (1, 0).

Learn how to determine the cosine of three pi over two using the unit circle, noting the point has x = 0, so cosine equals zero.

Compute the cosine of five pi over three using the unit circle and the known value cos pi over three equals 1/2; determine the sign by quadrant.

Compute the cosine of pi via the unit circle, noting that cosine is the x-coordinate and pi corresponds to the point (-1, 0).

learn how to compute cosine of pi over two using the unit circle; identify the point (0,1) on a unit circle and see that cos(pi/2) equals zero.

Compute the cotangent of five pi over six by writing it as cosine divided by sine; use cos(5π/6)=−√3/2 and sin(5π/6)=1/2 to obtain cot(5π/6)=−√3.

Compute the cosecant of 11 pi/6 by using the sine reciprocal, with the reference angle pi/6 in quadrant four on the unit circle, yielding negative two.

Compute the cosecant of 13 pi over three by hand by rewriting in terms of sine, using unit circle and reference angle to determine the sign, and rationalizing the result.

Compute cosine of two pi over three by hand, using secant as reciprocal and the reference angle pi over three. In quadrant II, the cosine sign is negative.

Compute the sign of zero using the unit circle and sine at angle zero. Confirm that the y coordinate is zero, so the sign of zero is zero.

Compute the sign of two pi by hand using the unit circle, identifying the point at two pi with coordinates (1, 0) and noting that sine equals zero.

We compute the sign of sin(3π/2) by sketching the unit circle. We locate the point (0, -1), and since sine equals the y coordinate, its sign is negative.

Compute the sign of pi by hand using the unit circle, locating the point at pi as (-1, 0). From the y-coordinate, deduce the sign is zero.

Compute the sign of pi/2 using the unit circle, noting that at pi/2 the point is (0,1) and sine equals 1.

Compute the tangent of three pi over four by rewriting as sine over cosine, locate the angle on the unit circle, and find tangent equals negative one.

Use the unit circle to find the cosine of two pi. The point at two pi has coordinates (1, 0), so cosine two pi equals 1.

Apply a minimal memorization method to determine the sine sign of five pi over three using the reference angle pi over three and the unit circle, yielding a negative value.

Use the reference angle pi over six to determine the sign of sine for seven pi over six. In quadrant three, sine is negative and equals one half.

solve for the y coordinate on the unit circle by substituting x = sqrt(3)/2 into x^2 + y^2 = 1, then find y = -1/2 in quadrant four.

Show that the point (√3/3, √6/3) lies on the unit circle by substituting into x^2 + y^2 = 1 and obtaining 1.

Solve a right triangle by using Pythagoras to find the missing side, then apply cosine to find angle a and use the triangle sum to get angle b.

Apply the tangent ratio to determine the shadow length from a 23.4° sun elevation: a 5.75 ft person yields about 13.3 ft.

Use the angle of elevation and the tower's 30-meter shadow; apply tangent to compute height as h = 30 tan(32.1°), giving about 18.8 meters.

Explore the sine and cosine functions: unit circle graphing, period two pi, amplitude and range, phase shift, vertical translation, and the standard forms for sine and cosine.

Graph y = 1/2 cos(1/2 x - pi/4) by mapping x to 0–2 pi, solving for x, dividing into four subintervals, and plotting points on the unit circle.

Graph sec(1/2 x) by graphing cosine on [0, 2π], scaling to [0, 4π], and plotting key ordered pairs, then draw asymptotes where cosine is zero and sketch the secant.

Graph the cosecant by graphing its reciprocal sine, with G(x) equals three halves sin(x - pi/2); compute delta x to subdivide 0 to 2pi, plot points, and identify asymptotes.

Solve 2x = -pi/2 and 2x = pi/2 to get x = -pi/4 and x = pi/4, the vertical asymptotes, then sketch the tangent curve.

Sketch the graph of y = (1/2) cot(2x) and identify its zeros and vertical asymptotes at x = 0 and x = pi/2.

Explore simple harmonic motion using s(t)=a cos(ωt) or s(t)=a sin(ωt), with amplitude |a|, period 2π/ω, and frequency ω/2π, illustrating oscillation about equilibrium in a spring-mass system and the maximum height.

Explore simple harmonic motion through a cosine height function, identifying amplitude four, period pi/5, and frequency five over pi, and find the first maximum at t equals pi/10 seconds.