Question

drag each label to the correct location. not all labels will be used. first, drag a value to represent the missing angle in the triangle. then, complete the trigonometry equality statements. \\(\sin(\theta)\\) \\(59^{\circ}\\) \\(90 + \theta\\) \\(57^{\circ}\\) \\(62^{\circ}\\) \\(90 + 33\\) \\(\cos(\theta)\\) \\(90 - \theta\\) \\(\sin(28^{\circ}) = \cos(\quad)\\) \\(\cos(33^{\circ}) = \sin(\quad)\\) \\(\cos(31^{\circ}) = \sin(\quad)\\) \\(\cos(90 - \theta) = \quad\\)

Explanation:

Step1: Find missing triangle angle

The triangle is right-angled at B, so $\angle A + \angle C = 90^\circ$. Given $\angle C = \theta$, $\angle A = 90^\circ - \theta$. From complementary angle logic, $\angle A = 90^\circ - 33^\circ = 57^\circ$ (matches provided label $57^\circ$).

Step2: Solve $\sin(28^\circ)=\cos(\square)$

Use co-function identity $\sin(x)=\cos(90^\circ-x)$.
$\sin(28^\circ)=\cos(90^\circ-28^\circ)=\cos(62^\circ)$

Step3: Solve $\cos(33^\circ)=\sin(\square)$

Use co-function identity $\cos(x)=\sin(90^\circ-x)$.
$\cos(33^\circ)=\sin(90^\circ-33^\circ)=\sin(57^\circ)$

Step4: Solve $\cos(31^\circ)=\sin(\square)$

Use co-function identity $\cos(x)=\sin(90^\circ-x)$.
$\cos(31^\circ)=\sin(90^\circ-31^\circ)=\sin(59^\circ)$

Step5: Solve $\cos(90-\theta)=\square$

Use co-function identity $\cos(90^\circ-x)=\sin(x)$.
$\cos(90-\theta)=\sin(\theta)$

Answer:

Missing angle in triangle: $57^\circ$
$\sin(28^\circ)=\cos(62^\circ)$
$\cos(33^\circ)=\sin(57^\circ)$
$\cos(31^\circ)=\sin(59^\circ)$
$\cos(90 - \theta)=\sin(\theta)$