\(\tan A=(-0.5-2)/3=-5/6\). The reference angle is \(\alpha=\tan^{-1}(5/6)=39.80557\ldots^\circ\). Thus \(A=180-\alpha=140.1944\ldots^\circ\) or \(360-\alpha=320.1944\ldots^\circ\). Adjacent angles have opposite tangents, so \(B=\alpha\) or \(180+\alpha\). The answers are \(39.8^\circ\) and \(219.8^\circ\).

\(y=\frac{\sqrt3}{2}\times\frac1{\sqrt3}=\boxed{\frac12}\).

\(\boxed{\cos45^\circ=\frac1{\sqrt2}}\) and \(\boxed{\tan60^\circ=\sqrt3}\).

The principal angle is \(\cos^{-1}(0.41)=65.795165\ldots^\circ\). The reflex solution is \(360^\circ-65.795165\ldots^\circ=\boxed{294.2^\circ}\).

(a) Draw a smooth sine wave through \((0^\circ,0),(90^\circ,1),(180^\circ,0),(270^\circ,-1),(360^\circ,0)\).

(b) \(\sin x=-4/5=-0.8\). The reference angle is \(53.130102\ldots^\circ\). Sine is negative in quadrants III and IV, giving \(x=180^\circ+53.130102\ldots^\circ\) or \(360^\circ-53.130102\ldots^\circ\). Hence \(\boxed{x=233.1^\circ\text{ or }306.9^\circ}\).

(a) Draw a smooth cosine curve through \((0^\circ,1),(90^\circ,0),(180^\circ,-1),(270^\circ,0),(360^\circ,1)\).

(b) \(\cos x=-1/3\). The principal angle is \(109.471220\ldots^\circ\); the other is \(360^\circ-109.471220\ldots^\circ\). Thus \(\boxed{x=109.5^\circ\text{ or }250.5^\circ}\).

\(\cos x=-2/5=-0.4\). The principal angle is \(113.578178\ldots^\circ\), and the second is \(360^\circ-113.578178\ldots^\circ\). Thus \(\boxed{x=113.6^\circ\text{ or }246.4^\circ}\).

(a) \(\boxed{\tan60^\circ=\sqrt3}\).

(b) \(\sin x=1/2\). The reference angle is \(30^\circ\); sine is positive in quadrants I and II. Thus \(\boxed{x=30^\circ\text{ or }150^\circ}\).

(a) Draw a smooth sine curve through (0°,0), (90°,1), (180°,0), (270°,−1), (360°,0).

(b) \(\sin x=-1/5=-0.2\). The reference angle is \(\sin^{-1}(0.2)=11.536959\ldots^\circ\). Sine is negative in quadrants III and IV, giving \(\boxed{x=191.5^\circ\text{ or }348.5^\circ}\).

\(\tan30^\circ=1/\sqrt3\). Tangent is negative in quadrants II and IV, so \(\boxed{x=150^\circ\text{ or }330^\circ}\).

(a) Draw a smooth curve through (0°,1), (90°,0), (180°,−1), (270°,0), (360°,1).

(b) \(\cos x=-\sqrt3/2\). The reference angle is 30° and cosine is negative in quadrants II and III, so \(\boxed{x=150^\circ\text{ or }210^\circ}\).

(a) Draw a smooth cosine curve through (0°,1), (90°,0), (180°,−1), (270°,0), (360°,1).

(b) The reference angle is 45°. A reflex angle with positive cosine lies in quadrant IV, giving \(\boxed{x=315^\circ}\).

(a) Draw a smooth cosine curve through (0°,1),(90°,0),(180°,−1),(270°,0),(360°,1).

(b) The reference angle is \(\cos^{-1}(0.294)=72.902417\ldots{}^\circ\). Cosine is positive in quadrants I and IV, giving \(\boxed{x=72.9^\circ\text{ or }287.1^\circ}\).

tan x=−√3; reference angle60°. Tangent is negative in quadrantsII andIV, so \(\boxed{x=120^\circ\text{ or }300^\circ}\).

\(\sin x=-5/8\). The reference angle is \(\sin^{-1}(5/8)\). Sine is negative in quadrants III and IV, giving \(\boxed{x=218.7^\circ\text{ or }321.3^\circ}\).

(a) (i) \(\boxed{\text{cubic}}\); (ii) \(\boxed{\text{reciprocal}}\). (b)(i) A smooth sine curve passes through (0,0), (90,1), (180,0), (270,−1), (360,0). (b)(ii) sin x=−0.4; reference angle=sin⁻¹0.4. The two solutions are \(\boxed{203.6^\circ,336.4^\circ}\).

\(\tan x=-2\); reference angle=\(\tan^{-1}2=63.4349\ldots^\circ\). Quadrants II and IV give \(\boxed{x=116.6^\circ\text{ or }296.6^\circ}\).

\(\tan x=-4/3\). The reference angle is \(\tan^{-1}(4/3)=53.130\ldots^\circ\). Tangent is negative in quadrants II and IV: \(\boxed{x=126.9^\circ\text{ or }306.9^\circ}\).

(a) Increasing branches, zeros at \(0^\circ,180^\circ,360^\circ\) and vertical asymptotes at \(90^\circ,270^\circ\).

(b) \(\tan30^\circ=1/\sqrt3\). With period \(180^\circ\), solutions \(30^\circ,210^\circ\).

(a) Draw a smooth cosine curve through \((0^\circ,1),(90^\circ,0),(180^\circ,-1),(270^\circ,0),(360^\circ,1)\).

(b) \(\cos x=-3/5\). \(x=126.9^\circ\) or \(360^\circ-126.8698976\ldots^\circ=233.1^\circ\).

Reference angle \(\sin^{-1}(3/5)=36.8698976\ldots^\circ\). Sine is negative in quadrants III and IV.

\(x=180+36.8698976\ldots=216.9^\circ\) or \(x=360-36.8698976\ldots=323.1^\circ\).

(a) Through \((0,1),(90,0),(180,-1),(270,0),(360,1)\).

(b) \(x=360^\circ-\cos^{-1}(0.21)=282.1^\circ\).

(a) At \(x=45,60,80\), the missing values are \(0,-1.5,-2.8\). (b) Draw a smooth curve through the points.

(c) The horizontal line \(y=-2\) meets the curve at about \(x=66\). Calculator check \(x=\frac12\cos^{-1}(-2/3)=65.9\).

(d) Divide by 40 to get \(y=2-x/40\). Draw this line, e.g. through \((0,2),(80,0)\). Its intersection with the curve gives \(x\approx34\) (numerical check 33.64).

(a) The sine curve passes through \((0,0),(90,1),(180,0),(270,-1),(360,0)\).

(b) \(\sin x=-1/3\). Reference angle \(\sin^{-1}(1/3)=19.4712^\circ\). Solutions are \(x=180+19.4712=199.5\)° and \(x=360-19.4712=340.5\)°.

\(\sin x=-2/3\). Reference angle \(\alpha=\sin^{-1}(2/3)=41.8103^\circ\). Solutions \(180+\alpha=221.8\)° and \(360-\alpha=318.2\)°.

(a) Cosine passes through \((0,1),(90,0),(180,-1),(270,0),(360,1)\).

(b) Reference angle 60°. Cosine negative in quadrants II and III: \(x=120,240\)°.

(a) Sine passes through \((0,0),(90,1),(180,0),(270,-1),(360,0)\).

(b) \(3-2\sin x=13/4\) gives \(\sin x=-1/8\). Reference angle \(\sin^{-1}(1/8)=7.1808\ldots\)°. Hence \(x=187.2,352.8\)°.

(a) The sine curve passes through (0°, 0), (90°, 1), (180°, 0), (270°, −1) and (360°, 0).

(b) \(\sin x=-.8\). Reference angle \(\alpha=\sin^{-1}(.8)=53.1301\ldots^\circ\). Negative sine occurs in quadrants III and IV: \(x=180^\circ+\alpha=233.1^\circ\) or \(x=360^\circ-\alpha=306.9^\circ\).

\(\sin x=.75\). Reference angle \(\sin^{-1}(.75)=48.6^\circ\). Sine is positive in quadrants I and II, giving 48.6° and \(180-48.6=131.4^\circ\).

(a) Increasing tangent branches, zeros at 0°, 180°, 360° and vertical asymptotes at 90°, 270°. (b) \(x=\tan^{-1}(1/5)=11.3^\circ\). Tangent has period 180°, so the other solution is \(191.3^\circ\).

\(\cos x=5/3-2=-1/3\). The solutions are \(x=109.5^\circ,\ 250.5^\circ\).

\(\sin x=-2/7\). Reference angle \(\sin^{-1}(2/7)=16.6^\circ\). Sine is negative in quadrants III and IV: \(x=196.6^\circ,\ 343.4^\circ\).

Reference angle 60°. Tangent is positive in quadrants I and III: \(x=60^\circ,240^\circ\).

(a) Smooth cosine curve through (0°, 1), (90°, 0), (180°, −1), (270°, 0), (360°, 1). (b) \(\cos x=1/4\), so \(x=\cos^{-1}(1/4)=75.5^\circ\) or \(x=360^\circ-75.522\ldots^\circ=284.5^\circ\).

(a)(i) Smooth sine curve through (0°,0), (90°,1), (180°,0), (270°,−1), (360°,0). (ii) Rotational symmetry of order 2 about (180°,0).

(b) \(\sin x=3/4\). The two roots are \(x=\sin^{-1}(.75)=48.6^\circ,\quad180^\circ-48.6^\circ=131.4^\circ\).

(c)(i) \((x+5)^2-11\). (ii) Upward parabola with minimum (−5,−11), y-intercept14 and roots \(-5\pm\sqrt{11}\).

Tangent is positive in quadrantsI andIII. \(x=\tan^{-1}2=63.4^\circ\) or \(63.4^\circ+180^\circ=243.4^\circ\).

Tangent is negative in quadrantsII andIV. Reference angle \(\alpha=\tan^{-1}(4/3)=53.130\ldots^\circ\). Thus \(x=180^\circ-\alpha=126.9^\circ\) or \(360^\circ-\alpha=306.9^\circ\).

(a) \(x=\sin^{-1}(.36)=21.1^\circ\). (b) \(180-21.100196\ldots=158.9^\circ\).

\(x=180-\sin^{-1}(.43)=154.532\ldots\approx154.5^\circ\).

\(\tan100^\circ<\cos100^\circ<1/100<100^{-0.1}\). Approximate values −5.671, −0.174, 0.01, 0.631.

The values are approximately 0.375, −0.927, 0.788 and 0.616. Thus \(\cos38^\circ>\sin38^\circ>\sin158^\circ>\cos158^\circ\).

\(\sin120^\circ=\sqrt3/2\), so \(3\sqrt3/2-4(3\sqrt3/8)=0\).

(a) The curves meet at x = 45°. (b) Find where the sine curve is 0.5 above the cosine curve: x ≈ 66°. For verification, \(\sqrt2\sin(x-45^\circ)=0.5\) gives \(x=45^\circ+\sin^{-1}(1/(2\sqrt2))\approx65.7^\circ\) in the displayed interval.