Questions: (a) Find an angle θ, with 0°<θ<360°, that has the same cosine as 53° (but is not 53° ). θ= i .53 (b) Find an angle θ, with 0°<θ<360°, that has the same sine as 53° (but is not 53° ). θ= i

(a) Find an angle θ, with 0°<θ<360°, that has the same cosine as 53° (but is not 53° ).
θ= i .53
(b) Find an angle θ, with 0°<θ<360°, that has the same sine as 53° (but is not 53° ).
θ= i

Transcript text: (a) Find an angle $\theta$, with $0^{\circ}<\theta<360^{\circ}$, that has the same cosine as $53^{\circ}$ (but is not $53^{\circ}$ ). $\theta=$ i .53 $\square$ (b) Find an angle $\theta$, with $0^{\circ}<\theta<360^{\circ}$, that has the same sine as $53^{\circ}$ (but is not $53^{\circ}$ ). \[ \theta=\mathbf{i} \]

Solution

Solution Steps

To solve these problems, we need to use the properties of trigonometric functions.

(a) For cosine, the cosine of an angle is the same as the cosine of its supplement. Therefore, if we have an angle $ \theta = 53^\circ $, the other angle with the same cosine in the range $ 0^\circ < \theta < 360^\circ $ is $ 360^\circ - 53^\circ $.

(b) For sine, the sine of an angle is the same as the sine of its supplement. Therefore, if we have an angle $ \theta = 53^\circ $, the other angle with the same sine in the range $ 0^\circ < \theta < 360^\circ $ is $ 180^\circ - 53^\circ $.

Step 1: Finding the Angle with the Same Cosine

To find an angle $ \theta $ such that $ 0^{\circ} < \theta < 360^{\circ} $ and $ \cos(\theta) = \cos(53^{\circ}) $ (but $ \theta \neq 53^{\circ} $), we use the property of cosine. The angle that satisfies this condition is given by: \[ \theta = 360^{\circ} - 53^{\circ} = 307^{\circ} \]

Step 2: Finding the Angle with the Same Sine

To find an angle $ \theta $ such that $ 0^{\circ} < \theta < 360^{\circ} $ and $ \sin(\theta) = \sin(53^{\circ}) $ (but $ \theta \neq 53^{\circ} $), we use the property of sine. The angle that satisfies this condition is given by: \[ \theta = 180^{\circ} - 53^{\circ} = 127^{\circ} \]