Trigonometry 2 with The Math Sorcerer

Learn to verify trigonometric identities by starting with the more complex side, rewriting with sine and cosine, and transforming to secant and cosecant forms to demonstrate equality.

Verify a trigonometric identity by starting with the more complex side, rewriting cotangent and tangent in sine and cosine, and using cosine squared theta plus sine squared theta equals one.

Verify the identity by starting with the left-hand side, perform the distribution, substitute sec^2 theta = 1 + tan^2 theta, and obtain tan^2 theta as the right-hand side.

Verify a trig identity by converting cos^2 x csc x minus csc x to -sin x. Use sine-cosine relations and the cos^2 x + sin^2 x = 1 identity.

In this video solution to problem 4, verify a trig identity by rewriting tangent in terms of sine and cosine and using cos^2 x plus sin^2 x = 1.

Verify a trigonometric identity by rewriting the left-hand side cos^2 x minus tan^2 x over sin^2 x as cot^2 x minus sec^2 x, using reciprocal relationships and division rules.

Find cos 15 degrees by rewriting 15 as 45 minus 30 and applying cosine of a minus b identity, yielding (sqrt(6) + sqrt(2)) / 4.

Use cosine sum and difference identities to compute the exact value of cos(5π/12). Decompose 5π/12 into π/6 + π/4, apply the identity, and simplify to (√6 − √2)/4.

Apply the cosine addition identity to compute cos(87°)cos(93°) - sin(87°)sin(93°); conclude cos(180°) = -1 using the unit circle.

Apply the cosine difference identity, substitute x=180 and y=θ, use unit circle values cos180 = -1 and sin180 = 0, conclude cos(180°−θ) = −cosθ.

Compute the exact value of sin 75 degrees by rewriting it as 45 plus 30 and applying the sin(X+Y) identity, yielding (sqrt6+sqrt2)/4.

Compute the exact value of tan(pi/12) by writing pi/12 as pi/3 minus pi/4 and applying the tangent difference formula, then rationalize the denominator for a simplified result.

apply the cosine sum identity to express cos(3π/2 + x) as sin x, by substituting a and b with 3π/2 and x, and using the unit circle values.

Write pi/12 as pi/3 minus pi/4 and apply the sin difference identity. Compute sin(pi/12) = (√6 - √2)/4.

Use the tangent addition identity to express tan(7pi/12) as tan(pi/3+pi/4) with tan(pi/3)=the square root of three and tan(pi/4)=1, rationalizing the denominator to obtain the negative square root of three.

Use the sine difference identity sin x cos y − cos x sin y with x = 40° and y = 160° to get sin(40°−160°). This equals sin(−120°) = −sin(120°) = −√3/2.

Apply the double-angle identity sin(2x)=2 sin x cos x to simplify four times sin 15° cos 15°. This yields 2 sin 30° = 1.

Apply the identity 2 cos^2 x minus 1 equals cos 2x, divide by two, substitute pi/8, and obtain the exact value: the square root of two over four.

Factor out two and use cos 2x = 1 - 2 sin^2 x with x = 15°, giving 2 cos 30° = sqrt(3) as the exact value.

Apply cos 2x = cos^2 x − sin^2 x with x = 15 degrees to obtain cos 30 degrees. Cos 30 degrees equals the squared of three over two.

Apply the double-angle identity cos 2x = 1 − 2 sin^2 x to rewrite 2 − 4 sin^2 15° as 2(1 − 2 sin^2 15°), then simplify to 2 cos 30° and obtain sqrt(3).

Apply the sine half-angle identity to sin(-pi/8), determine the sign from quadrant four, substitute cos(-pi/4) = sqrt(2)/2, and simplify to the exact value.

Use the half-angle identity to find cos(x/2) when cos x = 1/4 and x is in the first quadrant. Therefore cos(x/2) = sqrt(5/8), with the positive sign.

Use half-angle identities to find cos(x/2) given sin x = -4/5 with x in the fourth quadrant, deducing cos x = 3/5, and obtain cos(x/2) = -2/√5 = -2√5/5.

Apply the cos difference identity to convert cos 4x minus cos 2x into -2 sin(3x) sin(x) by substituting x with 4x and y with 2x.

Apply the sine difference identity to rewrite sin x minus sin y as 2 cos((x+y)/2) sin((x-y)/2); plug x=102 and y=95 to obtain 98.5° and 3.5°.

Apply the cos x cos y identity to 5 cos 3x cos 2x, using x+y and x-y to reduce to cosine of x.

Apply sin 2x = 2 sin x cos x to rewrite sin x cos x as 1/2 sin 2x; substitute x = 15°, use sin 30° = 1/2, giving 1/4.

Apply the double-angle identity cos 2x = 1 - 2 sin^2 x with x = 22.5°, yielding cos 45° = sqrt(2)/2.

Apply the cosine half-number identity to compute cos(pi/12) by using the plus sign for quadrant I and expressing pi/12 as half of pi/6, then simplify to a radical form.

Learn how inverse trig functions—arc sine, arc cosine, and tangent inverse—map from output back to input, with their respective ranges and domains, and practice undoing functions.

Apply the cosine inverse to cos(-pi/6) using the even nature of cosine, recall cos(pi/6) = sqrt(3)/2, and conclude y = pi/6 with y in [0, pi].

Compute cos theta where theta = tan^{-1}(-2) using a right triangle with opposite -2 and adjacent 1; the hypotenuse is sqrt(5), so cos theta = 1/sqrt(5) = sqrt(5)/5.

Apply the cosine double-angle identity to cos(2 theta) where theta = arctan(4/3). Build a 3-4-5 triangle to get cos theta and sin theta, then compute cos(2 theta) = -7/25.

Use the unit circle to show cos(3π/2)=0, so cosine inverse of zero equals π/2 and the range of cosine inverse is [0, π], avoiding naive cancellation.

Explore how to evaluate the arcsin of the sine of 3π/2 using the unit circle, and identify that arcsin(-1) equals -π/2 within the inverse sine range of -π/2 to π/2.

Apply the double-angle tangent identity to tan(2 theta) with theta = cos^-1(1/4), using a right triangle to find tan theta = sqrt(15) and obtain tan(2 theta) = - sqrt(15)/7.

Relate theta to u via arcsin, use toa to form a right triangle, solve for x with Pythagoras (x = sqrt(1 - u^2)), and express cotangent as adjacent over opposite.

Define theta as arctan(4x) and form a right triangle with opposite 4x and adjacent 1. Use the pythagorean theorem to get hypotenuse sqrt(1+16x^2), then secant of theta equals sqrt(1+16x^2).

Example 9 demonstrates solving for the missing side of a right triangle by setting theta to the inside expression, using arc sine, applying Pythagoras, and yielding cos(arcsin((A-B)/C)) = sqrt(C^2-(A-B)^2)/C.

Use the inverse sine definition and memorize the arcsin range [-pi/2, pi/2] to find the exact value. On the unit circle, sin y = sqrt(2)/2 yields y = pi/4.

Rewrite cosine inverse of minus one half as cosine, apply the inverse cosine range 0 to pi, and locate the angle in quadrant two with reference angle pi/3, yielding 2pi/3.

Let theta be arc cosine of one fourth; use cos theta = adjacent/hypotenuse with adjacent=1 and hypotenuse=4, then opposite = sqrt(15) and sin theta = sqrt(15)/4.

Solve -2 cos x + 1 = 0 on [0, 2pi], yielding x = pi/3 and x = 5pi/3, and determine where f(x) is negative or positive across these intervals.

Isolate sine x to solve equation, yielding sin x = -1/2, then use unit circle with reference angle π/6 to obtain x = 7π/6 and 11π/6 in 0 to 2π.

Isolate the sine function to solve the trigonometric equation and show sin x = 6/5 exceeds the range of -1 to 1, so there is no solution, i.e., empty set.

Solve a trigonometric equation with x in [0, 2pi] by isolating tangent to get tan x = -1, yielding x = 3pi/4 and 7pi/4 from sine and cosine signs.

Isolate the cosine and solve 3 cos x = -5 to get cos x = -5/3, then conclude there is no solution, the empty set.

Learn to solve trick trig equations by substituting u = x/2 and equating sine u with cosine u, yielding x = pi over two as the unique solution.

Use cos^2 x − sin^2 x = cos 2x to solve cos 2x = 1 for x in 0 to 2pi, yielding x = 0 and pi.

Apply double-angle identity to transform sin x cos x into sin 2x, then solve sin u = 1/2 with u = 2x to obtain x = pi/12, 5pi/12, 13pi/12, 17pi/12.

solve for all x with csc(2x) defined by sin(2x)=1/2, using the unit circle; obtain x=pi/12 + k pi and x=5pi/12 + k pi

Solve sin x = 1/2, obtaining base angles pi/6 and 5pi/6. With period 2pi, all solutions are x = pi/6 + 2k pi or x = 5pi/6 + 2k pi.

Rewrite cot x as cos x over sin x in sin x cot x minus sin x, cancel sin x, then cos x = sin x; solutions π/4 and 5π/4.

Factor 2 sin^2 x + 3 sin x + 1 = 0 to obtain sin x = -1/2 or sin x = -1; thus x equals 7π/6, 11π/6, or 3π/2.

Solve 2 cos x + 1 = 0 on [0, pi], obtain cos x = -1/2, identify angles with reference angle pi/3, yielding x = 2 pi/3 and pi/3.

Factor sin^2 x + sin x - 6 = 0 into (sin x + 3)(sin x - 2); sin x = -3 or 2 has no solution.

Factor the equation into (cos x + 1)^2, solve cos x = -1 on 0 to pi, and use the unit circle to get x = pi.