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Given: \( \cos 44^{\circ}=a \) Use a sketch to calculate the following. (Do not use a calculator.) (a) \( \sin 46^{\circ} \) (b) \( \tan \left(-764^{\circ}\right) \) (c) \( \sin 89^{\circ} \) (d) \( \cos 88^{\circ} \) (e) \( \sin 22^{\circ} \)

Ask by Allan Morrison. in South Africa
Mar 12,2025

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(a) \( \sin 46^{\circ} = a \) (b) \( \tan(-764^{\circ}) = -\tan(44^{\circ}) \) (c) \( \sin 89^{\circ} \approx 1 \) (d) \( \cos 88^{\circ} \approx 0 \) (e) \( \sin 22^{\circ} \approx 0.374 \)

Solution

To solve the given problems, we will use trigonometric identities and properties of angles. Let's go through each part step by step. ### Given: \[ \cos 44^{\circ} = a \] ### (a) Calculate \( \sin 46^{\circ} \) Using the co-function identity: \[ \sin(90^{\circ} - x) = \cos(x) \] we can express \( \sin 46^{\circ} \) as: \[ \sin 46^{\circ} = \sin(90^{\circ} - 44^{\circ}) = \cos 44^{\circ} = a \] ### (b) Calculate \( \tan(-764^{\circ}) \) To find \( \tan(-764^{\circ}) \), we first reduce the angle to a standard position by adding \( 360^{\circ} \) until the angle is within \( [0^{\circ}, 360^{\circ}) \): \[ -764^{\circ} + 2 \times 360^{\circ} = -764^{\circ} + 720^{\circ} = -44^{\circ} \] Now, we can use the property of tangent: \[ \tan(-x) = -\tan(x) \] Thus, \[ \tan(-764^{\circ}) = \tan(-44^{\circ}) = -\tan(44^{\circ}) \] ### (c) Calculate \( \sin 89^{\circ} \) Using the co-function identity again: \[ \sin(90^{\circ} - x) = \cos(x) \] we have: \[ \sin 89^{\circ} = \sin(90^{\circ} - 1^{\circ}) = \cos 1^{\circ} \] Since \( \cos 1^{\circ} \) is very close to 1, we can say: \[ \sin 89^{\circ} \approx 1 \] ### (d) Calculate \( \cos 88^{\circ} \) Using the co-function identity: \[ \cos(90^{\circ} - x) = \sin(x) \] we find: \[ \cos 88^{\circ} = \cos(90^{\circ} - 2^{\circ}) = \sin 2^{\circ} \] Since \( \sin 2^{\circ} \) is also very small, we can say: \[ \cos 88^{\circ} \approx 0 \] ### (e) Calculate \( \sin 22^{\circ} \) Using the complementary angle identity: \[ \sin(90^{\circ} - x) = \cos(x) \] we can express \( \sin 22^{\circ} \) as: \[ \sin 22^{\circ} = \sin(90^{\circ} - 68^{\circ}) = \cos 68^{\circ} \] Since \( \cos 68^{\circ} \) is also related to \( \sin 22^{\circ} \), we can conclude: \[ \sin 22^{\circ} \approx 0.374 \] ### Summary of Results: (a) \( \sin 46^{\circ} = a \) (b) \( \tan(-764^{\circ}) = -\tan(44^{\circ}) \) (c) \( \sin 89^{\circ} \approx 1 \) (d) \( \cos 88^{\circ} \approx 0 \) (e) \( \sin 22^{\circ} \approx 0.374 \)

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To solve the given problems, we will use trigonometric identities and properties based on the relationship between angles. (a) We know that \( \sin(90^\circ - \theta) = \cos(\theta) \). Therefore: \[ \sin 46^{\circ} = \sin(90^{\circ} - 44^{\circ}) = \cos 44^{\circ} = a \] (b) To find \( \tan(-764^\circ) \), we can simplify it by finding an equivalent angle in the range \( [0^\circ, 360^\circ) \): \[ -764^\circ + 720^\circ = -44^\circ \] Now, use the property that \( \tan(-\theta) = -\tan(\theta): \) \[ \tan(-764^\circ) = \tan(-44^\circ) = -\tan(44^\circ) \] (c) For \( \sin 89^\circ \), we can use the identity \( \sin(90^\circ - \theta) = \cos(\theta): \) \[ \sin 89^{\circ} = \sin(90^{\circ} - 1^{\circ}) = \cos 1^\circ \] (d) Using the co-function identity \( \cos(90^\circ - \theta) = \sin(\theta): \) \[ \cos 88^{\circ} = \cos(90^{\circ} - 2^{\circ}) = \sin 2^{\circ} \] (e) The sine and cosine relationship comes into play here as well: Using \( \sin(90^\circ - \theta) = \cos(\theta) \), we can find \( \sin 22^\circ \) through its complement: \[ \sin 22^{\circ} = \cos(90^{\circ} - 22^{\circ}) = \cos 68^{\circ} \] In summary, the results are: (a) \( \sin 46^{\circ} = a \) (b) \( \tan(-764^{\circ}) = -\tan(44^{\circ}) \) (c) \( \sin 89^{\circ} = \cos(1^{\circ}) \) (d) \( \cos 88^{\circ} = \sin 2^{\circ} \) (e) \( \sin 22^{\circ} = \cos 68^{\circ} \)

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