Trigonometry › Exact trigonometric values
Exact values Solve the first problem 18 problems
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Cosine of 15 degrees ●○○○○ 10 ptsFind the value of cos 15 ∘ \cos 15^\circ cos 1 5 ∘ Double-angle sine from given sine and cosine ●○○○○ 10 ptsIf θ \theta θ sin ( θ ) = 3 5 \sin(\theta) = \frac{3}{5} sin ( θ ) = 5 3 cos ( θ ) = 4 5 \cos(\theta) = \frac{4}{5} cos ( θ ) = 5 4 sin ( 2 θ ) \sin(2\theta) sin ( 2 θ ) Cubed sine at half angle ●○○○○ 10 ptsIf θ = 60 ∘ \theta = 60^\circ θ = 6 0 ∘ 4 sin 3 θ 2 4\sin^3\frac{\theta}{2} 4 sin 3 2 θ Arcsine of sine of 10 ●●○○○ 20 ptsEvaluate sin − 1 ( sin 10 ) \sin^{-1}(\sin 10) sin − 1 ( sin 10 ) 10 10 10 Ratio of sine and cosine differences ●●○○○ 20 ptsEvaluate sin 35 ∘ − sin 25 ∘ cos 35 ∘ − cos 25 ∘ \frac{\sin 35^{\circ} -\sin 25^{\circ}}{\cos 35^{\circ} -\cos 25^{\circ}} c o s 3 5 ∘ − c o s 2 5 ∘ s i n 3 5 ∘ − s i n 2 5 ∘ Sine of a double angle from a shifted sine ●●○○○ 20 ptsGiven α ∈ ( 0 , π 2 ) \alpha \in (0, \frac{\pi}{2}) α ∈ ( 0 , 2 π ) sin ( 2 α − π 3 ) = 3 3 \sin (2\alpha - \frac{\pi}{3}) = \frac{\sqrt{3}}{3} sin ( 2 α − 3 π ) = 3 3 sin 2 α \sin 2\alpha sin 2 α Sine of a shifted double angle ●●●○○ 50 ptsGiven f ( x ) = 2 cos ( x − π 3 ) cos x + 1 f(x)=2\cos \left(x-\frac{\pi}{3}\right)\cos x+1 f ( x ) = 2 cos ( x − 3 π ) cos x + 1 f ( α + π 12 ) = 7 6 f\left(\alpha +\frac{\pi}{12}\right)=\frac{7}{6} f ( α + 12 π ) = 6 7 α ∈ ( 0 , π 2 ) \alpha \in \left(0,\frac{\pi}{2}\right) α ∈ ( 0 , 2 π ) sin ( 7 π 6 − 2 α ) \sin \left(\frac{7\pi}{6}-2\alpha \right) sin ( 6 7 π − 2 α ) Sine of a difference from given values ●●●○○ 50 ptsGiven that α , β ∈ ( π 3 , 5 π 6 ) \alpha, \beta \in \left(\frac{\pi}{3}, \frac{5\pi}{6}\right) α , β ∈ ( 3 π , 6 5 π ) sin ( α + π 6 ) = 4 5 \sin\left(\alpha + \frac{\pi}{6}\right) = \frac{4}{5} sin ( α + 6 π ) = 5 4 cos ( β − 5 π 6 ) = 5 13 \cos\left(\beta - \frac{5\pi}{6}\right) = \frac{5}{13} cos ( β − 6 5 π ) = 13 5 sin ( α − β ) \sin(\alpha - \beta) sin ( α − β ) More in Exact trigonometric values Negative angles and reciprocal ratios Tangent values