## ​ The Value of Sine, Cosine and Tangent for Angle θ,0∘≤θ≤360∘

 6.1 The Value of Sine, Cosine and Tangent for Angle $$\theta, 0^\circ \leq θ \leq 360^\circ$$

 What is corresponding reference angle? The corresponding reference angle, $$\alpha$$, is always less than $$90^\circ$$. Angles in quadrants II, III and IV have corresponding reference angles, . The angle in quadrant I itself is the corresponding reference angle, $$\alpha=\theta$$. The reference angles in quadrants II, III and IV are the corresponding angles in quadrant I.

 Example 1 Determine the quadrant and the corresponding reference angle for each of the following. a) $$138^\circ$$     (b) $$239^\circ$$      c) $$312^\circ$$ Solution: \begin{aligned} \text{a)}\hspace{1mm}&\text{138^\circ is located in quadrant II.}\\ &\text{Corresponding reference angle, } \alpha\\ &= 180^\circ - 138^\circ\\ &= 42^\circ \end{aligned} \begin{aligned} \text{b)}\hspace{1mm}&\text{239^\circ is located in quadrant III.}\\ &\text{Corresponding reference angle, } \alpha\\ &= 239^\circ - 180^\circ\\ &= 59^\circ \end{aligned} \begin{aligned} \text{c)}\hspace{1mm}&\text{312^\circ is located in quadrant IV.}\\ &\text{Corresponding reference angle, } \alpha\\ &= 360^\circ - 312^\circ\\ &= 48^\circ \end{aligned}

 The relationship between the function of sine, cosine and tangent for angles in quadrants II, III and IV with the corresponding reference angle Quadrant II \begin{aligned} &\sin \theta = + \sin \alpha = + \sin (180^\circ - \theta)\\ &\cos \theta = - \cos \alpha= - \cos (180^\circ - \theta)\\ &\tan \theta = - \tan \alpha = - \tan (180^\circ - \theta) \end{aligned} Quadrant III \begin{aligned} &\sin \theta = - \sin \alpha = -\sin ( \theta-180^\circ)\\ &\cos \theta = - \cos \alpha= - \cos (\theta-180^\circ)\\ &\tan \theta = - \tan \alpha = +\tan (\theta-80^\circ) \end{aligned} Quadrant IV \begin{aligned} &\sin \theta = - \sin \alpha = -\sin ( 360^\circ-\theta)\\ &\cos \theta = + \cos \alpha= + \cos (360^\circ-\theta)\\ &\tan \theta = - \tan \alpha = -\tan (360^\circ-\theta) \end{aligned} In summary:

 What is a unit circle? The diagram below shows a unit circle. A unit circle is a circle that has a radius of 1 unit and is centered on the origin. The $$x$$-axis and -axis divide the unit circle into 4 equal quadrants, namely quadrant I, quadrant II, quadrant III and quadrant IV. It is given that P is a point that moves along the circumference of the unit circle and q is the angle formed by the radius of the unit circle, OP, from the positive x-axis in an anticlockwise direction. It is found that (a) point P is in quadrant I when$$0^\circ\lt\theta\lt90^\circ$$,  (b) point P is in quadrant II when $$90^\circ\lt\theta\lt180^\circ$$, (c) point P is in quadrant III when $$180^\circ\lt\theta\lt270^\circ$$, (d) point P is in quadrant IV when $$270^\circ\lt\theta\lt360^\circ$$.

 Relationship between the values of sine, cosine and tangent with the values of x-coordinate and y-coordinate for angles in quadrants II, III and IV in a unit circle.

 Example 2 The following diagram shows a unit circle and angle $$\theta$$. Determine the values of $$\sin \theta, \cos \theta\hspace{1mm}\text{and}\hspace{1mm} \tan \theta$$. Solution: \begin{aligned} \sin \theta &= 0.8829\\ \cos \theta &= 0.4695\\ \tan \theta&=\frac{0.8829}{ 0.4695}\\ &=1 .8805 \end{aligned}

The value of sine, cosine  and tangent at angles $$30^\circ, 45^\circ\text{ and } 60^\circ$$

 $$30^\circ$$ $$60^\circ$$ $$45^\circ$$ $$\sin \theta$$ \begin{aligned} 1\over 2 \end{aligned} \begin{aligned} \sqrt{3}\over 2 \end{aligned} \begin{aligned} 1\over \sqrt{2} \end{aligned} $$\cos \theta$$ \begin{aligned} \sqrt{3}\over 2 \end{aligned} \begin{aligned} 1\over 2 \end{aligned} \begin{aligned} 1\over \sqrt{2} \end{aligned} $$\tan \theta$$ \begin{aligned} 1\over \sqrt{3} \end{aligned} \begin{aligned} \sqrt{3} \end{aligned} \begin{aligned} 1 \end{aligned}

 Example 3 Without using a scientific calculator, determine the value for each of the following based on the corresponding reference angle. a)  $$\cos 150^\circ$$     b) $$\tan 225^\circ$$     (c) $$\sin300^\circ$$ Solution: \begin{aligned} \text{a) }\cos 150^\circ&=-\cos(180^\circ-150^\circ)\\ &=-\cos30^\circ\\ &=-\frac{\sqrt{3}}{2} \end{aligned} \begin{aligned} \text{b) }\tan 225^\circ&=+\tan(225^\circ-180^\circ)\\ &=+\tan45^\circ\\ &=1 \end{aligned} \begin{aligned} \text{c) }\sin 300^\circ&=-\sin(260^\circ-300^\circ)\\ &=-\sin60^\circ\\ &=-\frac{\sqrt{3}}{2} \end{aligned}

 Example 4 a) Given that $$\sin \theta=0.6157 \text{ and }0^\circ\leq\theta\leq360^\circ$$, calculate the angle $$\theta$$. b) Given that $$\cos \theta=-0.4226 \text{ and }0^\circ\leq\theta\leq360^\circ$$ calculate the angle $$\theta$$. c) Given that $$\tan \theta=-1.4826 \text{ and }0^\circ\leq\theta\leq60^\circ$$ calculate the angle $$\theta$$. Solution: \begin{aligned} \text{a) }&\sin \theta =0.6157 \text{(+ sign}\implies \text{quadrant I or II}) \\ &\text{Corresponding angle}\\ &=\sin^{-1}0.6157\\ &=38^\circ.\\ &\therefore\theta=38^\circ \text{ or } 142^\circ \end{aligned} \begin{aligned} \text{b) }&\cos \theta =-0.4226 \text{(- sign}\implies \text{quadrant II or III}) \\ &\text{Corresponding angle}\\ &=\cos^{-1}-0.4226\\ &=65^\circ.\\ &\therefore\theta=115^\circ \text{ or } 245^\circ \end{aligned} \begin{aligned} \text{c) }&\tan \theta =-1.4826 \text{(- sign}\implies \text{quadrant II or IV}) \\ &\text{Corresponding angle}\\ &=\tan^{-1}-1.4826\\ &=56^\circ.\\ &\therefore\theta=124^\circ \text{ or } 304^\circ \end{aligned}